Fracture Experiments of Single Grain Boundaries in Ductile ...

Fracture Experiments

of Single Grain Boundaries in Ductile

Metals

Vom Promotionsausschuss der

Technischen Universitat Hamburg-Harburg

zur Erlangung des akademischen Grades

Doktor-Ingenieur (Dr.-Ing.)

genehmigte Dissertation

von

Daniel Kupka

aus

Saarlouis

2013

Gutachter:

• Prof. Dr.-Ing. habil. Norbert Huber

• Prof. Dr. rer. nat. Gerold Schneider

Tag der mundlichen Prufung: 29. August 2013

urn:nbn:de:gbv:830-tubdok-12474

Fracture Experiments of Single Grain Boundaries in

Ductile Metals

Daniel Kupka

Institute of Materials Research, Materials Mechanics,

Helmholtz-Zentrum Geesthacht, 21502 Geesthacht, Germany.

Contact: [email protected]

Abstract

The fracture toughness of metals is significantly reduced when intergranular fractureis involved in the propagation process of a crack while the structure of the grainboundaries determines their resistance against fracture. Thus, relations betweenthe grain boundary structure and the macroscopic fracture toughness need to beestablished to improve predictions of the fracture process.

Within this work a test method was developed to fracture single grain boundariesin a plastically deforming metal. This method is based on a micro-cantilever bendingtechnique. The aluminum-lithium alloy 2198 was chosen as a model material dueto the morphology of the microstructure which provides flat elongated grains. Thegrains of this alloy were plastically deformed prior to the fracture of the grainboundary. However, the grain boundary showed brittle fracture; no damage wasfound prior to the complete fracture of the grain boundary.

The anisotropic plastic deformation properties of the grains were determinedusing a hybrid approach involving micro-column compression experiments on singlecrystalline columns and crystal plasticity finite element simulations. The adjustedmaterial model was applied within finite element simulations of the cantilevers andit was shown that the deformation of the cantilevers can be predicted by the finiteelement models.

Finally, the fracture of a single grain boundary was modeled making use of acohesive zone model. The methodology was applied to two cantilevers with sig-nificantly different deformation characteristics in order to determine the stress atdamage initiation of the respective grain boundaries.

I

Bruchversuche an einzelnen Korngrenzen in duktilen

Metallen

Daniel Kupka

Institut fur Werkstoffforschung, Werkstoffmechanik,

Helmholtz-Zentrum Geesthacht, 21502 Geesthacht, Deutschland.

Kontakt: [email protected]

Zusammenfassung

Metalle, bei denen sich Risse entlang von Korngrenzen ausbreiten, weisen im Allge-meinen eine geringere Bruchzahigkeit auf als Metalle bei denen transkristallinesRisswachstum dominiert. Die Trennfestigkeit der einzelnen Korngrenzen hangtwiederum von ihrer inneren Struktur ab. Daher mussen Beziehungen zwischender Struktur der Korngrenzen und der makroskopischen Bruchzahigkeit ermitteltwerden, um die Vorhersage des Bruchprozesses zu verbessern.

Im Rahmen dieser Arbeit wurde eine Methode entwickelt, die es ermoglicht,Bruchversuche an einzelnen Korngrenzen durchzufuhren. Die Methode basiert aufdem Biegen mikroskopisch kleiner Kragarmproben. Die Aluminium-Lithium Legier-ung 2198 wurde auf Grund der Gefugemorphologie, mit flachen Kornern, als Modell-material ausgewahlt. Bei den Biegeversuchen verformten dich die Korner zunachstplastisch, bevor der Bruch der Korngrenze einsetzte. Die Korngrenze selbst brachsprode, d. h. es konnte keine Schadigung der Korngrenze nachgewiesen werden bevordiese brach.

Die anisotropen plastischen Verformungseigenschaften der Korner wurden an-hand von Mikrokompressionsversuchen an einkristallinen Mikrosaulen bestimmt.Zur weiteren Auswertung wurden mit Hilfe dieser experimentellen Daten die Param-eter eines Finite Elemente Kristallplastizitatsmodells ermittelt. Es konnte gezeigtwerden, dass die Verformung der mikroskopischen Kragarme mit diesem kalibriertenMaterialmodell vorhergesagt werden kann.

Schließlich wurde der Bruch der Korngrenze unter Zuhilfenahme eines Kohasiv-zonenmodells modelliert. Die kritische Bruchspannung wurde exemplarisch fur zweiKorngrenzen in Proben mit deutlich unterschiedlichem Verformungsverhalten er-mittelt.

III

Contents

1 Introduction 1

2 Literature Review 42.1 Determination of Local Material Properties using Indentation Testing 42.2 Micro-column Compression Experiments . . . . . . . . . . . . . . . . 52.3 Micro-beam Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Micro-fracture Experiments . . . . . . . . . . . . . . . . . . . . . . . 72.5 Position of this Work in the Context of Micro-mechanical Testing . . 9

3 General Methods and Equipment 113.1 Imaging and Milling using a Focused Ion Beam Microscope . . . . . 113.2 Operating Principle of the Nanoindenter . . . . . . . . . . . . . . . . 133.3 Orientation Measurement by Electron Backscatter Diffraction (EBSD) 153.4 Digital Image Correlation (DIC) . . . . . . . . . . . . . . . . . . . . 16

4 Experimental Test Setup for Fracture Experiments on SingleGrain Boundaries 184.1 Choice of AA2198 as a Model Material . . . . . . . . . . . . . . . . . 184.2 Fabrication of Notched Cantilever Specimens . . . . . . . . . . . . . 204.3 Cantilever Bending Experiments . . . . . . . . . . . . . . . . . . . . 214.4 Corrections to the Experimental Data . . . . . . . . . . . . . . . . . 224.5 Methods for the Measurement of the Cantilever Stiffness . . . . . . . 264.6 Finite Element Framework for Normalization . . . . . . . . . . . . . 284.7 Experimental Evaluation of Damage Initiation and Propagation . . . 334.8 Morphology of the Crack Tip and the Crack Surface . . . . . . . . . 34

5 Determination of the Plastic Deformation Properties of the Grains405.1 Experimental Set-up and Data Analysis . . . . . . . . . . . . . . . . 405.2 Crystal Plasticity Model for Finite Element Simulations . . . . . . . 455.3 Experimental Determination of the Crystal Plasticity Parameters . . 49

6 Determination of the Grain Boundary Strength 586.1 Transfer of the Crystal Plasticity Model to the Cantilever Specimens 586.2 Cohesive Zone Model of the Grain Boundary . . . . . . . . . . . . . 636.3 Determination of the Initiation Stress for two Representative Can-

tilevers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

7 Conclusions 78

Bibliography 80

V

Nomenclature

A area of an indent

a depth of the U-notch

A average column area

B cantilever width

C11, C12, C44 elastic constants of a cubic symmetric material in Voigt notation

C0 . . . Cn constants describing the area function of the indenter tip

CRSS critical resolved shear stress

CSM continuous stiffness measurement

C fourth order elasticity tensor

d distance from the center of the U-notch to the support of thecantilever

D average column diameter

Dfoot diameter at the foot of the column

Dtop diameter at the top of the column

D damage variable

D,D∗,Dp total stretching tensor and stretching tensor associated with thelattice and the plastic deformation, respectively

E elastic modulus

e initial thickness of the cohesive layer

Eeff effective elastic modulus

Ei elastic modulus of the indenter material

Esim elastic modulus used within a simulation

FIB focused ion beam

Fp deformation gradient associated with plastic deformation

VII

F∗ deformation gradient associated with stretching and rotation

F deformation gradient

Gf fracture energy of the cohesive elements

H cantilever height

h displacement of the nanoindenter tip with reference to the samplesurface

h0 initial hardening modulus

H hardness

hcorconst correction for deviations from linear loading prior to the yieldpoint

hcorel displacement associated with elastic deformations only

h displacement rate of the indenter

hind indentation depth into a cantilever

hindel indentation depth corresponding to elastic deformations only

hindpl indentation depth corresponding to plastic deformations only

H0 initial height of a micro-column

hpl displacement of the nanoindenter tip associated with plastic de-formation only

hraw displacement of the indenter tip measured from the surface ofthe sample

hY displacement at the yield point

hαβ hardening moduli for the interaction of the deformed slip systemβ with the slip system α

K elastic cohesive normal stiffness

KCSMb elastic stiffness of the cantilever measured by CSM

KCSMsys combined stiffness of the cantilever and the indent measured by

CSM

Kexp cantilever stiffness determined in an experiment

K normalized stiffness

KS elastic cohesive shear stiffness

VIII

Ksim cantilever stiffness determined from a simulation

K∗ stiffness of a cantilever

L velocity gradient

L cantilever length

l characteristic length of a cantilever

Lp velocity gradient associated with plastic deformation

L∗ velocity gradient associated with stretching and rotation of thelattice

m Schmid factor for a slip system

mα slip direction of slip system α

nα normal vector of to the slip plane of slip system α

P load

P normalized load

Pmax maximum load on an indent

Pmax,i maximum load before unloading segment i

PY load at the yield point

qcp interaction coefficient for coplanar slip

qncp interaction coefficient for non-coplanar slip

qαβ hardening interaction coefficient for the interaction of slip sys-tems α and β

R radius of the U-notch

R rotation tensor

S indent stiffness

SCSMind indent stiffness measured by CSM

Sfc maximum loading slope in a micro-compression experiment

t cohesive normal traction

t cohesive normal traction if no damage would be present in thecohesive zone

ti cohesive normal traction at damage initiation

IX

vcrystal vector in the crystal coordinate system

vglobal vector in the global coordinate system

w cantilever deflection

wl cantilever deflection referring to a loading segment

wr cantilever deflection referring to a reloading segment

β geometrical factor related to the indenter tip geometry

δ cohesive displacement

δf cohesive displacement at total fracture

δi cohesive displacement at initiation of fracture

ϵ nominal strain rate of a micro-compression experiment

ε engineering plastic strain

εδ cohesive strain

γ accumulated plastic strain

γα shear rate on slip system α

γα0 reference shear rate on slip system α

ν Poisson ratio

νi Poisson ratio of the indenter material

ΩΩΩ,ΩΩΩ∗,ΩΩΩp total spin tensor and spin tensor associated with the lattice andthe plastic deformation, respectively

(ϕ1,Φ, ϕ2) Eulerian angles to describe a crystallographic orientation

σ engineering stress

σσσ Cauchy stress tensor

ς normal stress on a slip system

τ resolved shear stress on a slip system

τ0 initial resistance to slip

τα resolved shear stress on slip system α

τ∞ saturation hardening shear stress

ταY resistance against slip of slip system α at instant time

X

ταY rate of change in resistance to slip of slip system α

θ misorientaion angle

ξ ratio between the elastic moduli of AA2198 and pure aluminum

XI

1 Introduction

The resistance of metals to fracture is mostly characterized by macroscopic quanti-ties of which the fracture toughness is the most popular. Detailed knowledge of themicroscopic mechanisms involved during the fracture process is required to predictthe macroscopic fracture properties. These microscopic mechanisms are controlledby microstructural features such as the grain size and the ductility of the grains,inclusions or pores and the grain boundaries.

The macroscopic fracture properties of aluminum lithium lightweight alloys aresignificantly influenced by the structure of the grain boundaries. It has been foundthat the macroscopic fracture toughness of such alloys is reduced when cracks ad-vance along the grain boundaries [1]. With advanced heat treatment the yield stresswas shown to increase while the fracture toughness decreased with respect to thequenched material taken as a reference (see figure 1.1). A reversion heat treatmentwas carried out to dissolve any precipitates in the grain interior while the grainboundary structure largely remained unaltered. The yield stress of the referencematerial was restored, indicating that all precipitates in the grain interior had beendissolved but the fracture toughness was significantly decreased compared to thereference material. The difference between the quenched (I) and the reverted mate-rial (II) was attributed to the different structure of the grain boundaries which werecovered by precipitates in the reverted material as can be seen from the transmis-sion electron images in figure 1.1. The results of [1] clearly show that the structureof the grain boundaries is responsible for the decreased fracture toughness and, asexpected, a large extent of intergranular fracture was observed when grain boundaryprecipitates were present.

The mechanisms responsible for the increasing sensitivity of the aluminum lithiumalloys toward intergranular fracture are subject to debate. Several approaches arebased on local strain concentrations on particles at the grain boundaries or due toprecipitate free zones adjacent to the grain boundaries [2]. Unfortunately, macro-scopic experiments provide limited information about the fracture mechanisms onsingle grain boundaries due to the unknown local stress state at the respective grainboundaries and plastic deformation in the adjacent grains. In order to overcomethese limitations experimental methods are required to conduct tests on specificgrain boundaries involving well defined stress fields. The ability to fabricate small-scale site-specific test specimens of known geometry together with high resolutionload-displacement measurements enable the development of such experiments which,in turn, provide a better understanding of the material properties on the macro-scopic scale.

The challenge for this work was to develop a method that is capable of fracturingsingle grain boundaries and to provide a measure for their resistance to damage and

1

1 Introduction

(a)

(b)

Figure 1.1 (a) Yield stress versus fracture toughness for different heat treatmentsof an aluminum lithium alloy (figure reproduced from [1]). I marksthe reference specimens as quenched and II marks the fully revertedspecimens. (b) Representative TEM images from grain boundaries ofthe specimens corresponding to the heat treatments I and II.

fracture. In particular, the analysis required the anisotropic plastic deformation ofthe grains adjacent to the boundary to be taken into account. Therefore, a hybridapproach of micro-mechanical testing and finite element analysis incorporating acrystal plasticity model has been chosen. This leads to the following structure ofthis thesis (see figure 1.2): In chapter 2 the experimental methods to access thematerial properties on the micro-scale are reviewed with a focus on the techniquesthat enable microscopic fracture experiments. The basic techniques required forthe fabrication and the testing of microscopic specimens and the characterization ofthe material are outlined in chapter 3. Chapter 4 is dedicated to the developmentof the experimental approach for microscopic fracture experiments on single grainboundaries including the measuring techniques for the initiation and propagationof an interfacial crack. The plastic deformation of the grains adjacent to the grain

2

Figure 1.2 Key aspects of the thesis corresponding to chapters 4, 5 and 6.

boundary must be taken into account through an adequate constitutive law. Micro-column compression experiments on single crystals that provide access to the plasticdeformation of a single grain are presented in chapter 5. The plastic deformationis thereby expressed through a finite element crystal plasticity model. In chapter 6this model is transferred to the microscopic bending/fracture experiments. In orderto model the fracture of a grain boundary a cohesive zone, which allows to determinethe stress at damage initiation, is included into the simulations. Finally, the methodis evaluated by applying it to two cantilevers with significantly different deformationbehavior. In figure 1.2 the key aspects of this work as presented in chapters 4, 5and 6 are summarized. Finally, the conclusions that can be drawn from this workand suggestions for future work are presented in chapter 7.

3

2 Literature Review

During the last decade, the commercial availability of focused ion beam systemshas triggered the development of a wide variety of small scale specimens for novelmicro-mechanical testing, since it provides the possibility to fabricate specimensof various geometries at distinct locations in bulk materials with a precision ofapproximately ten nanometers. Together with highly precise measuring and ac-tuation techniques, experiments have been designed that provide insight into themicro-mechanical mechanisms that determine material properties regarding the de-formation and failure. Single components of the microstructure, e. g. single grains orlayers in composite materials, have been isolated and tested under well defined stressand strain fields. The results provide a better understanding for the microscopicprocesses which determine the material properties on the macroscopic scale.

The experimental methods that are presented here provide an overview of thepossibilities and limitations of micro-mechanical testing for studying local materialproperties with a focus on metals. In particular, microscopic fracture experimentson bulk materials and interfaces are presented which, to date, are limited to brittlematerials.

2.1 Determination of Local Material Properties usingIndentation Testing

The principle of instrumented indentation is to press an indenter of well definedgeometry into a surface while measuring the load and the displacement. The inter-pretation of indentation data to obtain material properties is described by Oliverand Pharr [3, 4]. A short review of this important method is given in chapter 3.The plastic deformability of the materials is usually described by the hardness. Themethod has been applied to a wide variety of materials including biological materi-als [5, 6], ceramics [7, 8], metals [9] and polymers [10]. Enabling measurements atsmall indentation depths, nanoindentation is particularly useful for investigationsinto the material properties of thin films [11, 12, 13].

The continuous measurement of load and displacement enables the design ofexperiments to determine elastic and plastic material properties [14, 15]. Moreover,they allow for the local determination of the material properties while the lateralresolution depends on the size of the indent and, thus, the indentation depth. Therehas been considerable effort in order to increase the lateral resolution by conductingmeasurements at small indentation depths. Today nanoindentation systems (ornanoindenters) have been developed which provide accurate control of load and

4

2.2 Micro-column Compression Experiments

displacement on the order of sub-nanometers and micro-Newtons, respectively. Theworking principle of a nanoindenter is described in chapter 3.

Indentation experiments provide the advantage of a relatively simple prepara-tion of the samples which makes the method quite versatile for a large variety ofspecimens and materials. However, the interpretation of the data from indentationexperiments is difficult due to the complex stress field under the indent and becausethe deformed volume constantly increases with increasing indentation depth.

2.2 Micro-column Compression Experiments

In order to overcome the difficulties associated with the interpretation of nanoin-dentation experiments, it is preferred to conduct tests with well defined uniaxialstress fields and constant volumes of deformation. Tensile and compression experi-ments on microscopic specimens meet these requirements. However, the challengesare to fabricate those specimens and to develop testing procedures that provide therequired high resolution in terms of load and displacement. The development of thefocused ion beam microscope (FIB) with high resolution milling capabilities (seechapter 3) has triggered the development of specimens with characteristic dimen-sions of less than 10 µm, while the high resolution of the nanoindenter in terms ofloads and displacement has been used to carry out the mechanical testing [16].

Tensile tests of microscopic samples for thin films [17, 18, 19] and bulk materi-als [20] have been developed. Such tests provide the desired uniaxial stress stateand constant deformation volumes. However, there are problems associated withthe fabrication, the handling and the loading of such specimens. In contrast, mi-croscopic compression tests can be conducted at any location within the materialand, moreover, the effort for the preparation and the handling of these specimensis significantly reduced.

In order to investigate the influence of the size of the specimens on the defor-mation properties Uchic et al. [16] fabricated cylindrical micro-columns into flatsurfaces of nickel single crystals. The yield stress of the columns was found to in-crease with decreasing diameter of the columns. Compression tests on single crystalsof metals have confirmed these findings and the yield stress has been found to beapproximately proportional to the inverse square root of the diameter [21, 22, 23].Various models have been proposed to explain this size-effect [24, 25, 26], but todate no consensus has been reached.

There are some limiting considerations to the interpretation of micro-compressionexperiments as discussed by Zhang et al. [27], Kiener et al. [28] and Kirchlechner etal. [29]. They mainly address inhomogeneities of the stress field which are caused bythe geometry of the column or by the imposed boundary conditions during loading.It is pointed out that friction between the indenter head and the top of the column isunavoidable and the indenter shaft is designed to be stiff regarding lateral directions.Hence, the deformation at the top of the column is constrained and a multiaxialstress field is induced. Moreover, stress concentrations may be induced due toirregularities at the attachment of the specimen to its substrate. A tapered column

5

2 Literature Review

geometry induces a strain gradient due to the variation of the diameter along thecolumns axis thereby reducing the influence of the attachment to the substrate. Ingeneral, higher aspect ratios would be desirable to create a homogeneous uniaxialstress field but this is circumvented by buckling of such specimens.

Multiple constraints associated with the geometry of the specimen and the testmethod must be taken into account for the analysis of the load-displacement datafrom micro-compression experiments. If these constraints are considered micro-compression experiments provide local measurements of the material properties. Aspointed out in section 5.1, finite element simulations are a highly versatile tool toincorporate the constraints into the analysis of the load-displacement data [30].

2.3 Micro-beam Bending

For some investigations on the microscopic scale tensile stresses are of interest.While micro-tensile experiments can be difficult to achieve, as described above,micro-bending experiments have been developed as an alternative. Moreover, suchtests allow for the investigation of the deformation properties of materials subjectto a well defined strain gradient.

Weihs et al. [31] designed a cantilever deflection experiment to determine theelastic moduli and the yield strengths of thin gold films on SiO2 substrates usinga simple beam bending theory. Florando and Nix [32] investigated the propertiesof copper thin films on silicon substrates by a similar technique, making use ofa cantilever with in-plane triangular geometry in order to obtain a uniform stateof plane strain in the film. They have shown that it is possible to separate theproperties of the film from the properties of the substrate by using an average stressmodel which does not impose any constraints on the properties of the film, i. e. noconstitutive law is assumed.

In order to investigate the effect of well defined strain gradients on the deforma-tion of metals Stolken and Evans [33] developed a bending method for thin foils.Their goal was to determine the inherent length scale of the material when applyinga phenomenological law of strain gradient plasticity. Annealed nickel foils were bentover a small mandrel which enables the determination of the bending moment andthe deformation. Starting from a strain gradient plasticity equation, the bendingmoment was linked to the strain gradient and the intrinsic length scale for straingradient plasticity from bending was determined.

Single crystalline cantilever beams have been tested by Motz et al. [34] to in-vestigate the effects of strain gradients in single crystals. Here the FIB was usedto fabricate cantilever beams out of a bulk copper sample which were then bentmaking use of a nanoindenter. Again a size effect was revealed with an increaseof the yield stresses proportional to the reciprocal height of the cantilevers. Thiseffect is explained by a combination of strain gradient plasticity with a pile-up ofdislocations at the neutral plane. Demir et al. [35] generalized this approach byintroducing a mean-field breakdown theory.

6

2.4 Micro-fracture Experiments

For further analysis of micro-beam bending experiments the deformation of thesupport of the cantilever and the exact geometry must be taken into account. Finiteelement simulations have been applied to assess the deformation properties in termsof stresses and strains. Gong and Wilkinson [36, 37] fabricated cantilevers withtriangular cross-sections out of single crystals of titanium and titanium alloys. Thecantilevers were oriented to deform along the ⟨a⟩ prismatic slip systems. Finiteelement simulations were applied to determine the critical resolved shear stress(CRSS) for these slip systems by a fitting procedure. Apart from the size effectwhich was attributed to the dislocation pile-up at the neutral plane, the influence ofalloying elements forming a solid solution or second phases was determined. It hasbeen shown that by extrapolating the CRSS values to the bulk, reasonable valuesfor the yield stresses of the respective alloys can be calculated.

While strain gradients have been shown to significantly influence the deformationproperties of small scale specimens, the exact mechanisms that are responsible forthe change of the mechanical properties are subject to current research.

2.4 Micro-fracture Experiments

The advance of micro-mechanical experiments to determine the local elastic andplastic properties of materials has triggered the development of microscopic fractureexperiments to characterize the failure of small structures. Fracture experiments onspecimens with dimensions on the order of less than a hundred micrometers aregenerally restricted to predominantly brittle materials. In ductile materials theplastic zone extends through the whole specimen circumventing any fracture.

Microscopic silicon cantilevers have been tested for their fracture strength byWilson et al. [38]. They fabricated a set of micro-cantilevers out of silicon wafersusing lithographic methods and carried out bending tests making use of a pin and aprecision displacement table. In order to prevent slipping of the pin on the cantileverthey fabricated a hole into the cantilevers which ensured a well defined loading point.The stresses and strains at the point of fracture were determined by making use ofan anisotropic elastic finite element model. The fracture stress between 3.0GPaand 3.6GPa were reported. Referring to inconsistencies in these values with otherreported values, the quality of the surface preparation is pointed out to be a decisivefactor for micro-mechanical experiments.

Halford et al. [39] used focused ion beam milling to create cantilever speci-mens of γ-TiAl alloys. Fracture tests have been conducted with specimens orientedalong various directions of the lamellar morphology of these alloys. As expected,a much smaller fracture toughness was determined for interlamellar fracture thanfor translamellar fracture. Investigations into the fracture strength of thin passiva-tion films (silicon oxide and silicon nitride) have been carried out by Matoy et al.[40]. The cantilevers were fabricated combining photolithographic techniques andfocused ion beam milling. The height of the cantilevers was chosen equivalent tothe thickness of the film material while the substrate material was used as supportof the fixed end. Additionally, a notch was milled into the cantilever close to the

7

2 Literature Review

supported end under perpendicular ion beam condition. A nanoindenter with aspherical tip was used to deflect the cantilevers without any visible imprint. Thefracture stresses were determined from the load-displacement data using analyti-cal solutions and finite element simulations. Moreover, finite element simulationsprovided the geometry function for the notch in the cantilever for evaluation ofthe fracture toughness. The fracture stresses and the fracture toughnesses deter-mined by microscopic experiments were in good agreement with the expected values.However, finite element simulations of the complete structure with its support wereneeded to determine the elastic moduli correctly from the bending experiments.Iqbal et al. [41] conducted similar fracture experiments on varying orientations ofNiAl-crystals. The determined values of the fracture toughness of [1 0 0] and [1 1 0]oriented cantilevers were in good agreement with values determined from macro-scopic tests. Additional fracture experiments were carried out in situ in a SEMallowing for the direct observation of the fracture process. The micro-cantileversin these works have been notched to create a well defined location for the fracture.During the notching process the ion beam was perpendicular to the top surface ofthe cantilevers. Hence, it is unknown for the specimens if the ion damage at theground of the notch influenced the results of the measurements.

Similar fracture experiments with unnotched cantilevers have been carried outby Klusner et al. [42] on ultra fine grained hard metal cantilevers of different sizes.With reduced specimen size inherent material defects have been avoided such thatthe high intrinsic tensile strength of the material could be determined.

In many cases of material failure damage is initiated at the interfaces betweenmicro-structural components. Here it is of great interest to evaluate the separa-tion properties of such interfaces in terms of maximum stresses and energies for theseparation of the surfaces. Promising experimental approaches have been adoptedfrom micro-fracture experiments using micro cantilever specimens on homogeneousmaterials. An interface of interest needs to be placed close to the supported end ofthe cantilever with the normal of the interface aligned with the beam axis. Hirakataet al. [43] followed this approach for investigations into the adhesion properties ofa thin (200 nm) copper film on a silicon substrate. The film was covered by a sili-con nitride passivation film (500 nm). Using focused ion beam milling, a cantileverspecimen was produced with silicon at the support and the interface normal of theSi-Cu and Cu-SiN interfaces aligned with the cantilever axis. The cantilevers weredeflected within a TEM since the positioning of the sample is crucial at the sub-micrometer dimensions. The initiation of the crack at the interface was observedand the critical load was determined. Finite element calculations assisted in thecalculation of the stress fields at the interfaces while it was assumed that the Si andSiN deform elastically only. A Ramberg-Osgood law was assumed for the copperlayer. The parameters to this law were determined such that the load-deflectioncurve could be reproduced by the finite element simulation. In addition to the can-tilever bending experiments millimeter sized three point bending specimens with aninterfacial crack were fabricated of the same material. The critical stresses for bothof these experiments are in good agreement such that it was concluded that bothprocesses can be described by a single criterion. Zhao et al. [44] reinvestigated the

8

2.5 Position of this Work in the Context of Micro-mechanical Testing

results and concluded that the influence of the plastic deformation of the copperlayer can be neglected and that a purely elastic model of the copper layer suitablydescribes the deformation and the induced stress fields. Therefore, any nonlinear-ity in the load displacement data has been attributed to the interfacial propertiesonly. Furthermore, they introduce cohesive zone elements into the finite elementsimulations to determine the separation properties of the Si-Cu interface by fittingthe parameters to match the load-displacement curves.

The method developed by Hirakata [43] has been applied by Matoy et al. [45]to interfaces between silicon oxide and films of copper, tungsten and tungsten co-sputtered with titanium respectively. The load deflection curves for these experi-ments are linear to the point of fracture. This indicates that only elastic deforma-tions need to be considered. Analytical calculations and finite element simulationsare used to determine the energy release rates for the fracture of these interfacesand the results of both methods are in good agreement. This indicates the po-tential of this method to measure the separation properties of interfaces on themicroscopic scale. The lowest energy release rates (weakest interface) are found forcopper films. Higher energy release rates are found for tungsten films and tungstenfilms co-sputtered with titanium show the highest energy release rates. It is pointedout, that the precision of these measurements strongly depends on the defect sizesin particular on the cantilever top.

Armstrong et al. [46] presented a method to separate grain boundaries in metalswhich makes use of cantilevers of pentagonal cross-section with a grain boundaryplaced close to its supported end. The cantilevers were deflected until the desiredmaximum load is reached. Some bending experiments were carried out in a acid so-lution of potassium tetrathionate. While the cantilever deflection remained constantat the maximum load with no acid present, the deflection increased continuously forexperiments which were carried out in the acid solution. Post mortem SEM imagesof the cantilevers revealed that the failure was induced at the grain boundary as de-sired. Most recent work has been carried out on copper grain boundaries that havebeen embritteled with bismuth [47]. In this material system the grain boundarieseither do not fracture or they fracture without any indication of plasticity priorto fracture. Moreover, no systematic dependence on the orientation of the grainboundary could be identified.

2.5 Position of this Work in the Context ofMicro-mechanical Testing

As just presented, a variety of methods have been developed to investigate thedeformation and fracture properties of materials at the micro-meter scale. To datefracture experiments have been developed for brittle materials without significantplastic deformation in the vicinity of the fracture site.

Within the present work the application of micro-mechanical experiments forthe investigation of the fracture of single grain boundaries is further developed.A method for the determination of the initiation and propagation of damage at

9

2 Literature Review

the grain boundary is presented. Moreover, the anisotropic plastic deformationof the adjacent grains is taken into account by combining finite element crystalplasticity and micro-mechanical experiments. Micro-column compression tests areshown to allow the calibration of the material model independently from the fractureexperiments. Finally, a cohesive zone has been introduced into the model to providea fracture criterion for the grain boundary.

Summary of the Methods for Micro-mechanical Testing

• Instrumented indentation provides access to the local material properties whilethe analysis of the acquired data is complex due to the complex stress fieldunder the indentation body and the increasing test volume.

• Micro-compression experiments are used to test constant small volumes whichfacilitates the analysis of stress strain behavior while the fabrication of thespecimens is a limiting factor to the use of the method.

• Micro-bending experiments induce well defined strain gradients in small spec-imens. Moreover, such experiments have been used to overcome the difficultfabrication of microscopic tensile specimens to test microscopic materials intension.

• Size effects have been found for various, mostly crystalline, materials andmodels for these size effects have been presented. When the dimensions ofthe specimens becomes small, both the yield stress and the hardening ratemay increase (“smaller is stronger”). These effects need to be considered forinvestigations into the deformation of small specimens.

• Micro-fracture experiments have been successfully conducted on both amor-phous and crystalline brittle materials such that analytical approximationscould be used for the analysis.

• Micro-fracture experiments on interfaces have been carried out on brittle in-terfaces with brittle matrix components.

• The preparation of microscopic specimens is subject to limitations regardingthe damage induced by the preparation technique and the precision of thegeometry.

10

3 General Methods and Equipment

The fundamental methods and equipment of micro-mechanical testing that havebeen used within this thesis are described in the present chapter. Since the focusedion beam has been used to fabricate the test specimens the principles of ion beammilling are reviewed. For the mechanical testing of the specimens a nanoindentationsystem that provides high resolution for both the load and the displacement has beenused. Hence, the working principle of a nanoindenter is presented. The principlesof electron back scatter diffraction (EBSD) are also presented since it has beenused to identify the individual orientations of the grains of the micro-fracture andmicro-compression experiments. Finally, the concept of digital image correlation ispresented, which has been used for the determination of local displacements duringdeformation.

3.1 Imaging and Milling using a Focused Ion BeamMicroscope

During the last decade the focused ion beam microscope (FIB) has evolved to anessential tool for the microscopical characterization of materials since it offers ca-pabilities for both imaging and micro-structuring [48]. The working principle issimilar to the working principle of a standard scanning electron microscope (SEM):ions are extracted from a source and accelerated by an electric field. Liquid metalion sources provide the highest brightness and due to the low melting temperatureand vapor pressure of gallium most commercially available FIB systems rely ongallium as source material. The ions energies are typically on the order of 5 eV to30 eV. Using an electrostatic lens system the beam is focused and deflection platesraster the beam over the surface.

The ions interact with the sample material, causing the emission of electronsand, moreover, sample material is removed due to the high mass of the incidentions. For imaging purposes usually the ion-induced secondary electron contrast isused. Crystalline materials show a strong dependence of the ion induced emissionof electrons on the crystal orientation and, thus, the ion imaging can be used foridentifying grains and grain boundaries. Moreover, the removal of sample materialfrom the surface provides the possibility to precisely machine microscopic structuresinto the surface. The rate of material removal also depends on the orientation ofthe crystal.

Imaging and milling procedures result in damage of the material due to incorpo-ration of incident ions, removal of material and local heating. The thickness of thedamaged layer depends on the sample material that is exposed to the ion beam and

11


the intensity of the ion beam. In particular, when micro-mechanical specimens arefabricated the direct exposure of the specimen to the ion beam needs to be limitedas far as possible. However, micro-mechanical specimens will be in contact with theincident ions at their side surfaces, which are parallel to the ion beam. The result-ing damage layers have a thickness of approximately 10 nm to 50 nm. Therefore,the side surfaces can be machined in several steps with subsequently reduced accel-eration voltage. This can significantly reduce the thickness of the damaged layer.For some materials also the chemical interactions need to be taken into accountsince the ion source material may interact with the sample and alter its properties,e. g. by alloying or by segregation. The influence of such a damaged layer on themechanical properties of the microscopic specimens is a current topic of scientificinvestigation [49].

Within this work the FIB has been used for the fabrication of microscopic speci-mens and for the identification of suitable locations for the fabrication of specimens.A system combining a SEM and a FIB (Nova Nanolab 200, FEI) allows for thepreparation and imaging of specimens. The FIB is located at an angle of 52 withrespect to the electron beam such that milling and in situ imaging without movingthe specimen is possible (see figure 3.1). The FIB and the electron beam coincide ata working distance (electron beam) of 5mm which is also the eucentric hight. Thesensitivity of the FIB to differences in the crystallographic orientation is used to im-age parts of the surface next to suitable positions for the specimen preparation whilethe location of the specimens are never directly exposed to the ion beam. Duringthe fabrication of the specimens, SEM images are used to adjust the milling pro-cess in order to obtain the desired specimen geometry. Since the specimen is neverexposed to the incident ion beam, no ion damage is introduced into the specimenswith exception of the damage layer at the surfaces of the specimens.

Figure 3.1 Schematic of a combined FIB/SEM system with a sample in the posi-tion of the standard milling procedure.

12

3.2 Operating Principle of the Nanoindenter

3.2 Operating Principle of the Nanoindenter

Instrumented indentation, or depth-sensing indentation, makes use of indenterswhich are pressed into the flat surface of the specimen while continuously mea-suring the load and the displacement. Making use of an analysis of the contactstresses that are present during the loading and unloading of the indenter, severalmaterial properties may be determined. For the analysis of the load-displacementdata at low indentation depths on the order of micrometers forces on the speci-mens must be exerted with a resolution on the order of micro newtons and thedisplacements must be measured with an accuracy in the order of nanometers. Ad-ditionally, lateral displacements normal to the indentation axis must be avoided. Asystem of three components meets all these requirements simultaneously (see fig-ure 3.2). The actuation is realized through a coil that is placed within an annularmagnet such that the electric current through the coil is directly proportional tothe applied load. Two leaf springs ensure that the applied load acts only in direc-tion of the indentation axis, in particular when contact is made. Finally, the highresolution displacement measurement is realized through a capacitive displacementgauge. The system is originally load controlled since the load is determined by thecurrent through the coil. Displacement control may be achieved through a feedbackloop. The phenomenon of thermal drift has to be taken into account, which meansthat the displacement may vary at a constant force due to thermal expansion orcontraction of the loaded assembly. In particular at low loading rates and highertemperatures this can affect the data significantly. To take the thermal drift intoaccount the displacement rate at a constant load can be measured after significantunloading of the specimen. This allows to take the effects of thermal drift intoaccount.

Figure 3.2 Schematic representation of a nanoindentation head.

The analysis of indentation experiments is based on contact mechanics and therehas been considerable development in understanding the load-displacement data [4].Here, only a brief introduction to the analysis of instrumented indentation withsharp indenters is given: Assuming that only elastic deformations are recoveredduring the unloading of the surface, the initial slope of the unloading curve is re-

13


garded as the elastic indentation stiffness, S. The elastic indentation stiffness isrelated to the contact area of the indent, A, by:

S = β2√πEeff

√A (3.1)

where β is a correction factor close to unity which is related to the shape of theindenter and Eeff is the effective elastic modulus that takes the deformation of theindentation body into account:

1

Eeff=

1− ν2

E+

1− ν2iEi

(3.2)

with the elastic modulus of the indentation body, Ei, the Poisson ratio of the in-denter, νi and the Poisson ratio of the material, ν. While the Ei and νi are known,the Poisson ratio of the material is mostly assumed; for metals the characteristicvalue of 0.3 is often a good estimation. In general, the contact area, A, may bedetermined by imaging the indents or by calculation from the contact depth of theindent, h, using the area function, fA(h):

fA(h) = A =

n=0∑8

Cn h1

2n−1

= C0h2 + C1h+ C2h

12 + C3h

14 + · · ·+ C8h

1128

(3.3)

For each tip the constants, C0 . . . C8, need to be determined through curve fittingof indentation data from indentation on materials with known material constants.Unfortunately, deviations from the contact area may occur when the material pilesup around the indent or when it sinks in. This alters the area function by reducing(“sink-in”) or extending (“pile-up”) the contact area. In particular in materials withlow work hardening capacity pile-up can cause problems in the analysis and needs tobe accounted for, e. g. by imaging the indents. The contact area, either measuredoptically or calculated for the indentation depth, may be used to determine thehardness, H, of the material which is defined as:

H =Pmax

A(3.4)

where Pmax is the maximum force exerted on the surface during the experiment.Some empirical relations between the hardness and the yield strength of particularmaterials (e. g. steels) have been established [50]. However, the hardness is nota fundamental material property but depends on the geometry of the indentationbody as well as on the elastic-plastic mechanical behavior of the tested material.

Moreover, dynamic indentation methods have been developed as an addition tothe standard methods [3]. During an indentation experiment a harmonic oscillationwith a small amplitude is superimposed to the indenter displacement. Comparingthe harmonic displacement to the harmonic load the stiffness of the indent may be

14

3.3 Orientation Measurement by Electron Backscatter Diffraction (EBSD)

determined continuously during an indentation experiment. For materials showingvisco-elastic/visco-plastic behavior (e. g. polymers) dynamic indentation methodsprovide a possibility for investigations into the time and rate dependent materialresponse [10].

As a final note, the indenter stiffness perpendicular to the indentation axis (lat-eral stiffness of the indenter) has a finite value. This can play an important role inthe experimental procedure when the lateral loads on the indenter are not balanced.This is the case for asymmetric loading due to inhomogeneities in the material oran asymmetric indenter tip. The lateral indenter stiffness depends on the displace-ment of the indenter shaft and is calibrated by the measurement of the lateral forcesversus the displacements; i. e. for each indentation depth the lateral stiffness of theindenter is known. Within the present work the lateral indenter stiffness must betaken into account for the cantilever bending/fracture experiments. During theseexperiments the location of the indent relative to the indentation axis moves due tothe deformation of the cantilever. The resistance to this movement must be includedin the analysis of the load-displacement data.

3.3 Orientation Measurement by Electron BackscatterDiffraction (EBSD)

When electrons are shot into a conductive material in a scanning electron microscopedifferent interactions can be found. A part of the electrons is scattered elastically,i. e. without loss of their energy. Since these electrons are scattered into all direc-tions, the source of the scattered electrons may be assumed to be a point sourcewithin the material. If the material is a crystal, the electron waves emitted fromsuch a source are diffracted by the lattice planes. Thereby, diffraction patterns areformed which may be used to identify the orientation of the crystals relative to aglobal coordinate system.

In order to use the back scattered and diffracted electrons for the local deter-mination of the crystal orientation the system described in the following has beenestablished (see figure 3.3). The energy of the electrons is typically in a range be-tween 10 eV and 30 eV. A phosphorous screen enables to capture the diffractionpattern of the backscattered electrons. The normal of the screen is placed at an an-gle of approximately 30 to the surface normal. This type of assembly allows for themeasurement in a domain of relatively high total intensities of the back scatteredelectrons. Making use of a CCD camera the diffraction patters on the phosphorousscreen can be captured.

The diffraction patterns form an assembly of bands on the phosphorous screen,where each band represents a lattice plane. The diffraction patterns are used toidentify the crystal orientation at the point of the incident electron beam. A Houghtransform [51, 52] is used to identify the positions of the bands and their intersectionson the screen. Knowing the crystal structure of the sample of interest the orientationof the crystal is determined by comparing the positions and intensities of the peaksin the Hough transform with the expected peaks of an ideal crystal. Using this

15


Figure 3.3 Schematic representation of an EBSD system.

information the orientation of the crystal relative to the sample coordinate systemas indicated in figure 3.3 is identified. Repeating this procedure at different pointsof the sample allows to create an orientation map.

3.4 Digital Image Correlation (DIC)

The measurement of local deformations in small dimensions requires techniqueswhich provide an adequate resolution of the displacement fields. The Digital ImageCorrelation (DIC) is capable of contactless capturing of such displacement fieldsfrom pixel based images, thus, without affecting the measurement in any way [53].The DIC algorithm takes two greyscale images as input: the reference picture andthe actual picture of the deformed structure. Characteristic subsets of pixels areidentified in both pictures and the displacement fields can be calculated from the rel-ative difference in positions. Hence, the native units of the measurement are pixelssuch that the DIC algorithm is scale independent. However, sufficiently contrastedimages providing distinguishable characteristic subsets of pixels are required to ap-ply the DIC algorithm. This is usually achieved by applying an irregular specklepattern if the surface of interest does not provide enough contrast.

In a first step, both images are divided into subsets of pixels which accuratelyrepresent the structure of the irregular speckle pattern. Making use of a bi-cubicinterpolation algorithm the images are smoothed, enabling for a sub-pixel resolu-tion in the order of 10−2 pixel. Bi-cubic interpolation is used since it generallyprovides a well-balanced relation between computation time and accuracy. The rel-ative displacement and the deformation of each subset are determined by finding thelocation of the subset in the actual image where the correlation coefficient reaches amaximum value. This correlation coefficient depends on the location, (x, y), of thesubset taken from the actual image and thus on the displacement of the center of

16

3.4 Digital Image Correlation (DIC)

the subset, (u, v). The calculation of the position of each point within the subsetadditionally requires the derivatives ∂u

∂x ,∂v∂y ,

∂u∂y and ∂v

∂x at the center of the sub-set. Various iteration methods may be used to determine the displacement and therespective derivations to find the maximum of the correlation coefficient, e. g. theNewton-Raphson method which provides good accuracy at acceptable computationtimes [54]. The software used within the present work is a free software availableat [55].

Making use of the DIC-method it is possible to determine two dimensional dis-placement and strain fields on a surface. The strain terms result from the derivationterms at each subset center. In order to apply the method for investigations intothe deformation of microstructural components it is necessary to create a sufficientlycontrasted surface. The problem to create a speckle pattern in this dimensions willbe addressed in chapter 6. Moreover, a suitable resolution of the images is requiredto achieve a sufficient accuracy to apply the DIC-method.

17

4 Experimental Test Setup forFracture Experiments on SingleGrain Boundaries

The investigation of the initiation and propagation of interfacial cracks at singlegrain boundaries requires the application of novel experimental techniques on themicroscopic scale. In this chapter the experimental challenges for quantitative frac-ture experiments of individual grain boundaries are specified. First, the aluminum-lithium alloy AA2198 is introduced as a model material for the development of amethod for separation of single grain boundaries. An experimental testing proto-col is established together with a data analysis procedure which necessitates theintroduction of corrections. Finally, a finite element framework is established thatprovides the possibility of comparing specimens with differences in their geometry.

4.1 Choice of AA2198 as a Model Material

The aluminum-lithium alloy AA2198 was chosen as a model material for the devel-opment of a single grain boundary separation method. Aluminum-lithium alloys areinteresting materials for investigations into the separation properties of single grainboundaries since intergranular fracture is known to reduce the fracture resistanceof these alloys [1]. An understanding of the mechanisms that lead to intergranularfracture is fundamental for the prediction of the macroscopic fracture properties.The concept of strain localization at or in the vicinity of the grain boundariesprovides an explanation for the susceptibility of these such alloys to intergranularfracture [2, 56, 57]. Severe strain localizations may be caused by precipitate freezones adjacent to the grain boundaries as well as by precipitates on the grain bound-aries. Such strain localizations lead to the nucleation and growth of voids whichfinally cause failure of the grain boundary. It has been shown that the structureof the grain boundaries determines the fracture mode between transgranular andintergranular failure rather than the ductility of the matrix [58, 59].

As described in section 4.2, focused ion beam (FIB) milling with a gallium liquidmetal ion source has been used for the preparation of the specimens. Since gallium isknown to embrittle grain boundaries in aluminum and its alloys, the analysis of themicro-mechanic processes that cause the failure of the grain boundaries is limited.However, the understanding of fracture processes which involve the combination ofbrittle grain boundaries and ductile grains provides an important challenge.

Rolled sheets of the material were provided by ALCAN in the temper T351 whichis also called “underaged”. A heat treatment to obtain the temper T8 was carried

18

4.1 Choice of AA2198 as a Model Material

out on some of the sheets; this temper is also called “peak-aged”. The macroscopicyield stress and the ultimate tensile strength have been determined from tensiletests in the rolling and the transversal direction. The results of these tests arelisted in table 4.1. The peak-aged material shows a higher yield stress and ultimatetensile strength, as expected. Additionally, nanoindentation experiments with aBerkovich indenter have been carried out to determine the microscopic hardnessof the alloy. The results averaged over 25 indentations for each heat treatmentare listed in table 4.2. The increase of the hardness of the peak-aged materialcompared to the hardness of the under-aged material is in good agreement withthe increase of the yield stress and the ultimate tensile strength measured from thetensile experiments. Cavaliere and de Santis [60] determined the elastic modulusfrom tensile tests as 74.3GPa for the transversal direction and 76.7GPa for thelongitudinal direction. However, other values of the elastic modulus have beenreported in the range between 71GPa [61] and 78GPa [62]. The actual elasticmodulus of the material is needed for the crystal plasticity model used in chapters 5and 6. However, the elastic deformation plays a minor role for these experimentssuch that it is reasonable to assume an elastic modulus of 75GPa for the followinginvestigations.

temper yield stress ,MPa UTS ,MPa

T351-L 324 445T8-L 470 511

T351-T 294 416T8-T 449 490

Table 4.1 Yield stress and ultimate tensile stress of the aluminum lithium alloyAA2198 from macroscopic tensile tests.

temper Hardness ,GPa

T351-T 1.322± 0.041T8-T 1.602± 0.069

Table 4.2 Hardness of the aluminum lithium alloy AA2198 measured by Berkovichindentation.

The microstructure of the material was revealed by polishing and subsequentetching, using Barker’s etch and observation under polarized light (see figure 4.1).The polishing procedure is described in table 4.3. The grains are elongated inthe rolling direction and flat respective to the thickness direction of the sheet. Inparticular, the grain boundaries are flat which is a great advantage for the bendingexperiments to determine the interfacial strength. In the rolling direction the lengthof the grains is generally larger than 300 µm while the grain size in the transversal

19

4 Experimental Test Setup for Fracture Experiments on Single Grain Boundaries

direction is between 100 and 300 µm. The thickness of the grains ranges between 1and 10 µm with an average thickness of about 3 µm.

Figure 4.1 Microstructure of the AA2198 sheets after Barker’s etch (HBF4+H2O,anodized using 0.2A/cm2), observed in polarized light.

Type Grain size Time

SiC 500 1minSiC 800 1minSiC 1000 1minSiC 1200 1minSiC 2500 1minDiamond 3 µm 3minOPS 0.05 µm 3min

Table 4.3 Polishing procedure for AA2198.

4.2 Fabrication of Notched Cantilever Specimens

As pointed out in chapter 2, micro-bending experiments on specimens with dimen-sions on the order of several micrometers have been carried out by various researchersto determine material properties of both thin films on a substrate and bulk materi-als. Furthermore, micro-cantilever specimens have been found to provide the mostpromising approach for microscopic fracture tests. Within this work a cantileverbending and fracture test has been established for investigations into the fractureproperties of single grain boundaries.

Cantilevers with a rectangular cross-section were prepared using FIB milling witha gallium ion source. A rectangular cross section was chosen in order to maintaina constant crack width. To achieve such a geometry, the specimens were fabricated

20

4.3 Cantilever Bending Experiments

from a near edge region of the sample in order to minimize the amount of materialto be removed by the FIB milling process. Both surfaces forming the edge had beenpolished prior to the fabrication process according to the procedure described intable 4.3. Using orientation contrast images achieved with the ion beam, a suitablegrain boundary was found and placed in a distance of approximately 2 µm fromthe supported end of the cantilever. Only the regions adjacent to the cantileverwere scanned such that the cantilever was never directly exposed to the ion beam.The selected boundary was marked by a line pattern outside the cutting area toidentify the position of the boundary during the cantilever fabrication process. Theorientations of the grains included in the cantilever were determined using the EBSDtechnique. In particular the orientations of the grains adjacent to the grain boundarywere important for the further analysis presented in chapter 6 since this is thelocation of the highest deformation and, moreover, the orientations can be used tocharacterize the grain boundary.

Using a relatively high beam current (e. g. 7 nA) a trench of material was re-moved from each side adjacent to the cantilever (figure 4.2(a)), leaving sufficientspace for subsequent cuts to release the cantilever from below (see figure 4.2(b)).Subsequent milling was conducted at lower currents in the range of 1 nA down to0.1 nA for the final surface milling. After the cantilever had been prepared, a U-notch was made using FIB milling from one side of the cantilever at the locationof the grain boundary. The notch ensures that the selected boundary serves as theprimary failure site during testing by avoiding high tensile stress concentrations inthe support. The blunt shape of the notch is favorable since the boundary edge maynot be perfectly straight, which would prevent the fabrication of a sharp artificialcrack directly at the boundary. Lastly, the well defined shape of the notch circum-vents any ambiguities associated the finite element analysis, as will be discussedlater. In the work presented here, the nominal dimensions of the final cantileverare approximately 20 µm× 5 µm× 5 µm (length×width× height). A representativecantilever is shown in figure 4.3.

4.3 Cantilever Bending Experiments

The bending experiments were carried out using a nanoindenter (Nanoindenter XP,Agilent) equipped with a Berkovich tip. This tip is used since it is sufficientlysharp to create visible indents on the cantilevers such that the bending length, L,of the cantilever from the supported end to the contact point for loading could bedetermined. Also, the lateral position of the indent could be identified for furtheranalysis. Moreover, the use of a sharp tip largely avoided the problems associatedwith a wedge or cylindrical contact, where misalignment at the contact would createtorque.

During the experiment, the cantilever was loaded at a constant displacementrate. A typical loading profile during an experiment is shown in figure 4.4. In orderto monitor the onset and propagation of damage, multiple partial unloads wereemployed for stiffness measurements; a decrease in the elastic stiffness was associated

21


(a) First milling process (b) Second milling process

Figure 4.2 Preparation of a cantilever from the edge of a sample.

Figure 4.3 SEM image of a cantilever with characteristic dimensions. The dashedline represents a grain boundary. L is the bending length, B is thecantilever width, H is the height of the cantilever, a is the depth of thenotch and d is the distance of the notch center to support.

with the presence of damage or crack initiation. Assuming small changes in geometrydue to plastic deformation, plasticity did not affect the unloading stiffness. Whileboth the unloading and reloading data may be used to determine the stiffness, herethe reloading data was used for the stiffness analysis. Additionally, the stiffnessof the cantilever was determined by the continuous stiffness method (CSM); thestiffness was measured by superimposing an oscillation with a small displacementamplitude of 2 nm to the loading procedure.

4.4 Corrections to the Experimental Data

While the applied load, P , is measured directly, the total displacement, hraw, is com-prised of the deflection of the cantilever beam, w, and the indentation depth, hind,as shown in 4.5(a). Figure 4.5(b) gives a schematic representation of the experiment

22


Figure 4.4 Load-displacement data from a typical micro-bending experiment.

illustrating the correction to be applied to the raw data. During the loading segmentthe indentation depth into the cantilever, hind(P ), is approximated by the indenta-tion depth associated with a standard indentation experiment conducted in the bulkmaterial at the same load. Hence, a reference curve was measured from indentationexperiments in the bulk material at loads up to the maximum load measured forthe bending experiments. The deflection of the cantilever alone during loading, wl,is expressed as:

wl(P ) = hraw(P )− hind(P ). (4.1)

It is worth noting that when the cantilever is deflected the indentation surface is nolonger normal to the indentation axis. However, the cantilever deflections presentedin this work do not exceed 10% of the bending length, leading to a maximuminclination of the cantilever-axis at the contact point of approximately 6. Theeffect of the inclination of the cantilever surface is neglected within the correction.

During the unloading and reloading segments the correction shown in equation(4.1) cannot be applied, due to partial elastic recovery of the indent. In orderto correct for the deflection of the cantilever during unloading and reloading theindentation depth must be decomposed into a plastic part, hindpl , and an elastic part,

hindel :hind = hindpl + hindel . (4.2)

During the un- and re-loading of the cantilever the plastic indentation depth is con-stant whereas the elastic indentation depth varies. The correction of the deflectionduring un- and re-loading is given by:

wr(P ) = hraw(P )− hindpl (Pmax,i)− hindel (P ), (4.3)

23


(a) (b)

Figure 4.5 (a) Schematic representations of the decomposition of the total mea-sured displacement, hraw, the cantilever deflection, w, the indentationdepth, hind, and (b) the correction of the raw data for the indentationdepth.

where the plastic indentation depth at the previous maximum is calculated as:

hindpl (Pmax,i) = hind(Pmax,i)− hindel (Pmax,i). (4.4)

In order to determine the elastic indentation depth, hindel (P ), the elastic indentationstiffness, S, of the material was determined from bulk indentation experiments. Theindentation stiffness is a function of the maximum indentation depth or equivalentlythe maximum applied load: S := S(P ). Either the use of multiple unloadingsegments or continuous stiffness measurements during indentation into the bulkcan provide the indentation stiffness as a function of the load, as needed for thecorrection. The elastic indentation depth is calculated as:

hindel (P ) =P

S(Pmax,i). (4.5)

The actual experimental stiffness of the cantilever, Kbeam, is calculated from theslope, dP

dw , of the corrected reloading segments.Finally, the elastic indentation depth also influences the dynamic measurement

of the stiffness by the CSM. In order to separate the continuous stiffness of thecantilever, KCSM

beam, from the stiffness of the indent, SCSMindent, the stiffness of the system,

KCSMsys , is modeled by two springs connected in series:

1

KCSMsys

=1

SCSMind

+1

KCSMb

. (4.6)

By rearranging (4.6) we find an expression for the continuous stiffness of the can-tilever alone:

KCSMb (P ) =

SCSMind (P ) ·KCSM

sys (P )

SCSMind (P )−KCSM

sys (P ). (4.7)

24


In order to determine the harmonic indent stiffness as a function of the applied load,indentation experiments using the CSM method were carried out in the bulk, up tothe maximum load used in the bending experiments. In particular, at low indenta-tion depths (hind < 100 nm) the variation of the elastic indentation depth cannot beneglected. Without considering the change in hindel , the slope of the unloading andreloading curves would be smaller than associated with the beam alone; leading toinaccurate stiffness calculations. The errors in the stiffness associated with the firstreloading segments would be particularly large since the elastic indentation stiffnessis lowest at small depths. It is worth noting, that the stiffness of the cantileveralone as calculated from equation (4.7) tends to infinity when SCSM

ind (P ) approachesKCSM

sys (P ). However, a typical stiffness measured in an experiment KCSMsys (P ) is

3000N/m. This value of SCSMind (P ) is achieved at a load of approximately 5 µN.

Therefore, this case has no practical impact on the measurements.

Calibration Curves for the Indentation Depth Correction

The correct determination of the cantilever deflection requires the indentation depthinto the cantilever as a function of the load, hind(P ). This correction function needsto be determined experimentally for each combination of indenter tip and material.For the cantilever bending experiments a Berkovich tip was used, with the correctionfactor β = 1.034. Typical loads in the cantilever bending experiments were on theorder of 2mN, leading to indentation depths of 200 nm with a Berkovich tip. Thecorresponding size of the indent was on the order of 1 µm. Since this is only 20% ofthe width of the cantilevers, it is reasonable to approximate the indentation depthinto the cantilever by evaluating the indentation depth into the bulk material at lowloads.

The load-displacement curves for the indentation experiments were establishedup to loads of 2.5mN which are the highest loads measured during the bendingexperiments. Five indentation experiments were carried out for each temper. Thedisplacement rate for these experiments was set to 10 nm/s. For a Berkovich tipthe area function is dominated by the square of the indentation depth. Therefore,assuming that the materials hardness is constant, the load-displacement data isexpected to follow a square function of the type:

P = a(hind

)2+ b hind. (4.8)

The linear term is introduced to account for deviations from the tip geometry froman ideally sharp tip. The constants a and b are determined from a fitting procedureof the load-displacement data for the whole set of experiments. Rearranging (4.8)provides the desired correction:

hind(P ) =

√P

a+

b2

4 a2− b

2 a. (4.9)

25


temper a, GPa b, kN/m c, mm2/N d, µm

T351 38.7 0.593 0.149 0.474T8 43.7 0.533 0.177 0.270

Table 4.4 Fit constants for the correction functions presented in equation (4.8) and(4.11).

The elastic stiffness of the indent also increases with increasing load. This correctionwas determined by indentation experiments in the bulk material, as well. The elasticindent stiffness, S(P ), was determined by making use of the continuous stiffnessmethod. Five measurements using CSM were carried out for each temper of thealloy. The maximum indentation load was 2.5mN and the displacement rate wasset to 10 nm/s.

Assuming that the hardness is constant at each indentation depth (3.4) resultsin:

A =P

H. (4.10)

Replacing the indent area, A, in equation (3.1) by this expression, a function of thesquare root type for the stiffness versus the indentation load is suggested. Using thesame method as described before, the indentation load, P , is fitted as a function ofthe elastic stiffness of the indent, S, assuming:

P = c S2 + d ·S, (4.11)

where the linear term is introduced to account for deviations of the indenter tip.However, the indentation stiffness versus load data is dominated by the quadraticterm. The correction is provided by rearranging (4.11):

S(P ) =

√P

c+

d2

4 c2− d

2 c. (4.12)

It is worth noting that the indent stiffness depends on the loading history, such thatthe maximum load applied to the indentation site determines the elastic stiffness ofthe indent. The elastic indent stiffness as a function of the depth is also used for thecorrection of the cantilever stiffness measured from the continuous stiffness methodby equation (4.7), i. e. SCSM

ind (P ) := S(P ).

4.5 Methods for the Measurement of the CantileverStiffness

In order to determine if the grain boundary of interest is subject to damage it isnecessary to measure the elastic stiffness of the cantilever; a loss of elastic stiffnessis associated with the occurrence of damage (e. g. [63]). By measuring the stiffnesscontinuously during the loading procedure the load and the displacement at the

26

4.5 Methods for the Measurement of the Cantilever Stiffness

(a)

(b)

Figure 4.6 Calibration curves for (a) the indentation depth and (b) the elasticindent stiffness versus the load.

27


point of initiation of damage can be determined and it is possible to observe thepropagation until complete failure.

In order to measure the elastic stiffness of the cantilever at each instant, theCSM method is favorable. However, it must be verified that the stiffness of thecantilever and the stiffness measured from the slope of the reloading segments are ofthe same value. For two of the specimens the indenter was unloaded every 200 nm ofdisplacement to 50% of the maximum load while a continuous stiffness measurementwas carried out. The harmonic frequency for the measurement was set to 45Hz andthe harmonic displacement was fixed at 2 nm.

The stiffness data is presented in figure 4.7 where both the harmonic stiffnessand the stiffness measured from the unloading segments are plotted against thecantilever deflection. The geometry of the cantilevers is given in table 4.5. The CSM-data shows scatter of approximately 15% around its average value. At deflectionsup to 100 nm the harmonic stiffness increases until it attains a constant averagevalue. This is likely due to deviations from the correction given in equation (4.7) atvery low indentation depths until full contact is established. The stiffness measuredfrom the reloading segments deviates from the mean value by approximately 5%,which shows that both methods are sufficiently accurate for the measurement ofthe cantilever stiffness. Figure 4.7(a) shows a specimen where the stiffness remainsapproximately constant until the point of failure. In figure 4.7(b) a specimen isshown where the stiffness decreases significantly before failure. Both methods wereable to capture this decrease of the cantilever stiffness. It is worth noting thatthe decrease of the stiffness, as shown in figure 4.7(b), is atypical for specimens ofboth tempers investigated in this work. As pointed out later, the grain boundariesmostly fail without significant loss of the structural stiffness prior to fracture. Thisis consistent with the expectation that the grain boundaries are embrittled due tothe gallium that is used during the fabrication process.

4.6 Finite Element Framework for Normalization

In order to compare the load-displacement data between specimens that have differ-ent geometries a normalization technique is required. Due to the complex geometryof the specimens involving the U-notch and the compliant support, finite elementmodeling was used. Making use of an isotropic linear elastic constitutive law, spec-imens with different geometries were compared in terms of the cantilever stiffnessand load.

For each specimen a finite element model was created to include the specificgeometry, as shown in figure 4.8. The main features of the geometry are: thebending length, L, defined from the supported end of the cantilever to the locationof the indent; the width, B; the height, H; the distance, d, from the supported endto the center of the notch; the notch depth, a; and the notch radius, R. Additionally,the cantilever support was explicitly modeled.

In order to create the FE-models, the finite element code ABAQUS was em-ployed. Eight-node brick elements were used to mesh the model except within the

28


(a)

(b)

Figure 4.7 Elastic cantilever stiffness measurements of cantilevers with (a) con-stant stiffness (CB-8) and (b) decreasing stiffness prior to fracture(CB-9). The stiffness measured from reloading segments (red dots),the elastic cantilever stiffness measured by CSM (blue curve) are givenfor the corrected datasets.

29


transition zone adjacent to the notch (see figure 4.8) where six-node wedge elementswere employed to reduce the mesh size. Fixed boundary conditions were imposed onthe surfaces at the bottom of the support as well as on its back surface. The elasticmodulus of the material at the loading point was 106GPa in order to prevent severedistortions of the elements. The resulting high contact stiffness negates the needfor any indentation correction in the finite element analysis; the displacement in thesimulation equals the cantilever deflection, w. The deflection in the simulation isgiven by a displacement boundary condition at the loading point.

Figure 4.8 Finite element model of a micro-cantilever. The geometric featuresgiven are the bending length, L, the width, B, the height, H, theposition of the notch center, d, the notch depth, a and the notch radius,R.

The indenter shaft provides a lateral stiffness which necessitates an additionalboundary condition in the finite element model to correctly describe the experiment.The lateral stiffness of the nanoindenter depends on the position of the indenter tiprelative to the load frame (see figure 4.9). For each experiment the lateral stiffnesswas introduced into the finite element model using two linear spring elements at theloading point.

For an estimation of the cantilever stiffness a linear elastic isotropic materialmodel was used. The elastic modulus in the simulation, Esim, is chosen as 75GPa,and a Poisson ratio of ν = 0.33 is assumed. Using these isotropic elastic constants,the stiffness, Ksim, of the structure was computed for each specimen. Assuming thatthe deflection is small compared to the bending length of the cantilever, a linear lawrelates the deflection, w, to the load, P , by a constant stiffness, K∗:

P = K∗w. (4.13)

All geometric features may be reduced to one single value with a dimension oflength which we call the characteristic length, l:

l =Ksim

Esim. (4.14)

30


Figure 4.9 Lateral stiffness of the indenter as a function of the position of theindenter tip relative to the load frame.

While the FE-model used here is for linear elastic behavior, it is not expectedthat the plasticity occurring during the actual experiments limits the applicabilityof the characteristic length for the normalization; in all cases the stiffness of interestresults from elastic deflections, and the permanent shape change resulting fromplasticity is small.

In the absence of damage we obtain:

P = Kexpwr = E l wr. (4.15)

A normalized load, P , and a normalized stiffness, K, are computed by dividingthe load and the stiffness by the characteristic length associated with the specimengeometry:

P =P

l(4.16)

K =Kexp

l. (4.17)

However, for a laterally constrained indenter the characteristic length depends onthe elastic modulus chosen for the simulation, Esim, as can be seen in figure 4.10.Therefore, the normalization approach may not be readily used to determine anunknown elastic modulus.

For the normalization linear elastic isotropic material behavior is assumed. Sinceoligo-crystalline cantilevers are tested, involving approximately ten grains in the de-formation process, the influence of the elastic anisotropy of the single grains hasbeen estimated. Simulations of single crystalline cantilevers with the crystallo-graphic axes [1 1 1] and [1 0 0] taken as the cantilever axis have been carried out.

31


Figure 4.10 Normalized stiffness as a function of the elastic modulus chosen forthe simulation for a laterally unconstrained and a laterally constrainedindenter.

For these simulations an anisotropic elastic material model with cubic symmetrywas applied. The cantilever with the [1 1 1] crystal orientation provides the highestelastic modulus in the direction of the cantilever axis, while the [1 0 0] orientationprovides the lowest elastic modulus.

Assuming that the anisotropy of the aluminum crystals is preserved the elasticconstants for the single crystal can be estimated. For pure aluminum the elasticconstants in Voigt notation are given as [64]: C11 = 108GPa, C12 = 61GPa andC44 = 29GPa. In order to determine a scaling factor, ξ, the isotropic elastic moduliof pure aluminum and AA2198 were compared to each other. The isotropic elasticmodulus of the alloy is taken to be 75GPa [60]. Pure aluminum has an isotropicelastic modulus of 70GPa. Hence ξ is calculated:

ξ =EAA2198

Epure Al= 1.071. (4.18)

The elastic constants for AA2198 are approximated as:

CAA219811 = ξ CAl

11 = 115.7GPa

CAA219812 = ξ CAl

12 = 65.3GPa

CAA219844 = ξ CAl

44 = 31.1GPa.

(4.19)

The simulations show that the deviations from the cantilever stiffness in a isotropicspecimen with the same geometry do not exceed 10%. These simulations of sin-gle crystals can be seen as a worst case where the elastic anisotropy of the single

32

4.7 Experimental Evaluation of Damage Initiation and Propagation

grains is most important. Since multiple grains are involved in the deformation ofthe specimens which have been investigated here elastic isotropy is assumed for theanalysis.

4.7 Experimental Evaluation of Damage Initiation andPropagation

SEM images were taken from each tested specimen to determine the location wherethe cantilevers were deformed, damaged or fractured. In most cases the cantileversfailed – as desired – at the chosen grain boundary as shown in figure 4.11. In afew cases the cantilevers failed at grain boundaries at the support or multiple grainboundaries failed during the test. Particularly stable grain boundaries or a badlypositioned notch can cause such a behavior when the stress to separate the grainboundary is not achieved. The analysis of the fracture parameters of the single grainboundary is considered in section 6.

Figure 4.11 Cantilever after the fracture test. The grain boundary under the U-notch failed as desired.

For cantilevers of each temper the development of the stiffness during the bendingexperiments was determined. The loading of the cantilevers was performed at aconstant loading rate of 5 nm/s. Every 100 nm of displacement the indenter wasunloaded to 50% of the maximum load for the determination of the stiffness. Theload-displacement data is then normalized through the procedure given in section 4.6including the lateral stiffness of the indenter in order to compare the cantilevers toeach other.

The experimental data of two cantilevers of each temper is shown here for com-parison. The geometry for of the cantilevers is given in table 4.5. The load versusdeflection data is shown in figure 4.12(a). The elastic stiffness, i. e. the slope ofthe reloading segments, differs significantly between the specimens due to differ-ences in the geometry. Making use of the FE-based normalization procedure thesedifferences are largely eliminated, as shown in figure 4.12(b). Furthermore, thenormalized stiffness is shown not to differ significantly between different specimens.

The normalized stiffness also refers to the elastic deformation of the specimen andcan be used to monitor the initiation and propagation of a crack. The accumulated

33


damage of the cantilever depends on the deformation history which is represented bythe maximum deflection of the cantilever at each instant. Hence, in order to comparethe evolution of the stiffness for different specimens, the slopes of the reloadingsegments are plotted against the deflection of the cantilever at the beginning of thecorresponding unloading segment. In figure 4.13 the cantilever stiffness is plottedfor specimens of both tempers. Generally, no significant decrease of the normalizedstiffness can be found with increasing deflection of the cantilever indicating brittleboundaries. As soon as the critical load or deflection for the initiation of a crackis attained, the grain boundary fails without stable crack propagation. Since nosignificant loss of elastic stiffness occurs prior to specimen failure, the nonlinearityin the normalized force is solely attributed to plastic deformation.∗

Specimen temper H µm L, µm B, µm a, µm d, µm

CB-1 T8 6.43 24.46 6.05 0.8 3.0CB-2 T8 8.89 24.8 5.14 0.64 4.71CB-3 T8 9.07 21.9 5.82 0.88 4.55CB-4 T8 9.43 25.23 5.57 1.05 3.20

CB-5 T351 6.48 17.43 7.11 1.01 2.13CB-6 T351 6.46 19.03 5.72 1.07 2.91CB-7 T351 6.84 18.40 4.31 1.09 1.28CB-8 T351 7.28 24.99 7.69 1.24 4.12CB-9 T351 8.33 23.18 5.81 1.6 5.90

Table 4.5 Geometry of the tested cantilever specimens. The geometric featuresgiven are the bending height, H, the bending length, L, the width, B,the notch depth, a and the position of the notch center, d. The notchradius for all specimens was 1 µm.

While it has already been stated that the plastic deformation of the specimendoes not affect the applicability of the linear elastic model for the normalizationof the load and the stiffness, this model cannot predict the stress fields in thecantilever at the point of failure. In order to overcome this limitation, the linearelastic constitutive law must be replaced by an appropriate constitutive model thatincludes plastic deformation. This is presented in chapter 5.

4.8 Morphology of the Crack Tip and the Crack Surface

In addition to the load-displacement response, the tested specimens were observedby high resolution electron microscopy. The crack tips and the fracture surfacesare of particular interest for studies of interfacial failure. Such observations provideinsight into the mechanisms of grain boundary failure. While the usefulness of a

∗It is worth noting that in very few cases a decrease of the stiffness is found prior to fracture.These samples have not been included for the further analysis presented in chapter 6.

34


(a)

(b)

Figure 4.12 (a) load-deflection curves and (b) corresponding normalized load-deflection curves for specimens of the tempers T351 (CB-6, CB-7)and T8 (CB-1, CB-2).

35


Figure 4.13 Normalized stiffness versus the deflection of the cantilever at the be-ginning of each corresponding unloading segment.

detailed analysis of the mechanisms involved in the failure of grain boundaries inAA2198 is limited to the effects of gallium embrittlement, several examples of thefeatures discernible from this technique are presented in the following paragraphs.

Figure 4.14 shows high resolution SEM-micrographs of the crack tips (figure 4.14(a)and 4.14(b)) or directly behind the crack tip (figure 4.14(c)) for various tested can-tilevers. Most crack tips were sharp irrespective of the thermo-mechanical condition,as shown in figure 4.14(a). In the grain on the left, slip traces are clearly visible.Such planar slip is known to cause stress concentrations at grain boundaries [65]which promote intergranular failure. Another feature which could be observed was adomain of bridging elements behind the crack tip also in an underaged specimen (seefigure 4.14(b)). Lastly, a blunted crack tip in a peak-aged specimen (figure 4.14(c))was observed. The interfacial crack in this specimen was stopped at a triple junc-tion, and massive slip occurred at the crack tip. The largely vertical lines which areseen in figure 4.14(b) and 4.14(c) result from the specimen fabrication through FIBmilling, the so-called “curtaining effect”, and should not be confused with slip lines.

For further investigations into the mechanisms involved in the initiation andpropagation of a crack, it is of great interest to examine the fracture surfaces aftertesting. The specimens remain connected to the support by a small ligament due tothe plastic deformation and compressive stresses on the lower part of the cantilever.It is also possible that the cantilever makes contact with the material or the tipof the indenter makes contact with the material surrounding the cantilever. Whilethe maintained attachment of the cantilever allows the identification of the indentercontact position which is needed to measure the bending length, the fracture surfacesare not readily accessible for imaging and need to be separated. In order to createenough space for the separation of the fracture surfaces using a Omniprobe micro-manipulator the cantilever was shortened using FIB milling. The micro-manipulatorwas carefully brought in contact with the shortened beam which was then deflected

36


(a) (b) (c)

Figure 4.14 (a) Sharp crack tip as observed in most specimens. Slip lines canbe seen on one of the adjacent grains. (b) Bridging elements foundbehind a crack tip of an underaged specimen. (c) Rounded crack tipin a peak-aged specimen showing massive slip on the crack ground.

to open the crack completely. It is worth noting that the complete removal ofthe cantilever up to the fracture is not possible since redeposition on the fracturesurfaces must be avoided.

After opening the cracks further, the fracture surfaces reveal additional infor-mation regarding the mechanisms of fracture. The underaged specimens exhibitgenerally smooth fracture surfaces with some larger particles visible, as shown infigure 4.15(a) and figure 4.15(b). Lines resembling fatigue striations which bowaround the coarse particles are found on the fracture surface (figure 4.15(a)). Fig-ure 4.15(b) shows the fracture surface of the specimen that exhibits the domain ofbridging elements shown in figure 4.14(b). The fracture surface may be divided intotwo regions. The first region resembles to the smooth surface of figure 4.15(a). Thesecond region shows a highly dimpled morphology. The fracture surfaces for thepeak-aged specimens show particles smaller than those found on the fracture sur-faces of the underaged specimens, as can be seen from figure 4.15(c). This fracturesurface corresponds to the crack tip shown in figure 4.14(c). The striation-like linesmentioned before are not found in any of the peak-aged specimens.

The features presented here give insight into the morphology of the intergranularcracks and the mechanisms involved in the fracture process. The coarse particlesfound on the fracture surfaces of the T351 specimens act as stress concentrators alsopromoting grain boundary failure. The multiple unloads used in the bending exper-iment can be one reason for the appearance of the striation-like lines on the fracturesurfaces of the T351 specimens since the number of unloads may be roughly corre-lated to the number of lines. However, other T351 specimens either do not showthese lines or the lines are less pronounced. The orientation of the boundary andthe adjacent grains and, in particular, the particles on the grain boundaries areexpected to influence the formation of these lines. From the fracture surfaces shownin figure 4.15(b) it can be seen that the fracture mode has changed from brittlefracture to dimpled fracture which indicates ductile deformation within the bound-

37


(a) (b)

(c)

Figure 4.15 (a) Smooth fracture surface of a grain boundary in an underaged spec-imen showing large particles and striation like lines. (b) Partially dim-pled fracture surface of an underaged specimen. (c) fracture surfaceof a grain boundary in a peak-aged specimen with finely distributedparticles, the arrow marks the position where the crack stopped.

ary. More investigations on the morphology of the grain boundary are required todetermine the cause for this change in fracture mode. However, due to the influenceof gallium on grain boundaries in aluminum a detailed study of the mechanisms ofgrain boundary failure is beyond the scope of this thesis.

Summary

• The aluminum alloy 2198 has been chosen as a model material since it providesa combination of ductile grains with brittle interfaces which is a fundamen-tal case for intergranular fracture in metals. The microstructure of this alloyprovides flat grain boundaries and therefore the possibility of creating micro-scopic bending specimens with a constant crack width. The grain boundaryproperties of this alloy may be altered by heat treatments.

38


• Additional embrittlement of the grain boundaries due to the FIB preparationinvolving a gallium ion source is acceptable for the development of the methodsince the material’s character of brittle interfaces and ductile grains is notchanged.

• An experimental method for the separation of single grain boundaries has beenestablished. Notched cantilever specimens with a rectangular cross-sectionhave been chosen since they provide the most promising approach for micro-scopic fracture experiments.

• Corrections to the experimental data that arise due to the use of a sharpindentation tip have been established.

• The stiffness measurements have shown that the specimens fail without signifi-cant propagation of the damage. The grain boundaries can, thus, be identifiedas brittle interfaces.

• A finite element framework for the analysis of the specimens has been estab-lished. This framework incorporating a linear elastic material can be used tocompare the load and the stiffness of similar specimens up to the yield loadand to identify the onset and propagation of damage. However, for the anal-ysis of stresses and strains at the moment of fracture, the plastic deformationof the specimens needs to be taken into account.

39

5 Determination of the PlasticDeformation Properties of theGrains

In order to describe the plastic deformation in the oligo-crystalline cantilevers theanisotropic deformation behavior of each grain must be taken into account. A fi-nite element crystal plasticity material model is capable to describe the anisotropicdeformation behavior. However, an independent determination of the plastic defor-mation properties is required. Since the determination of the model parameters isdifficult to perform from load-displacement data of polycrystalline structures andmacroscopic single crystals are not available for the specific temper of the alloy,micro-column compression experiments were carried out in order to provide uni-axial load-displacement data from single crystals of the alloy. The investigationsinto the plastic deformation behavior have been limited to the under-aged material(T351).

5.1 Experimental Set-up and Data Analysis

Micro-columns with a nominal diameter of 2 µm were prepared into the polished∗

surface of the sample. The transverse direction was parallel to the axis of the column,as shown in figure 4.1. Since the average grain thickness is 3 µm, the choice ofcolumns with a diameter of 2 µm allowed for the fabrication of single crystal columns.Combining EBSD-measurements and focused ion beam imaging, preferable grainsfor column fabrication were identified. Coarser grains were found from FIB images ofareas adjacent to the desired spot for milling. FIB marks were produced to identifythese locations during the later milling process. Then, the orientation of the grainsin these regions was determined by EBSD also revealing the shape of the grainssuch that the final milling location could be established. This location was markedby a “cross-hair” pattern in order to identify the position of the column axis usingFIB images without directly scanning over the column. Annular milling [21] wasused to fabricate the micro-columns, as schematically illustrated in figure 5.1. Theouter diameter was chosen to leave enough space for a flat punch indenter. Duringthe procedure concentric rings were milled into the surface while the ion currentwas stepwise reduced from 7nA to 50 pA. This enabled finer control of the columngeometry such that the taper angle could be reduced to less than 2. The nominaldimensions of the columns were 2 µm : 6 µm (diameter : height), resulting in an

∗see section 4.3 for polishing procedure

40


aspect ratio of 1 : 3. This aspect ratio provides a sufficiently homogeneous stressfield in the column while preventing instable deformation modes like buckling. Dueto the anisotropic milling rates for different crystallographic orientations the annularmilling procedure created an uneven surface. In order to reduce elevated differencesin the height of the surface, these differences were leveled after each milling step,but cannot be completely circumvented.

Figure 5.1 Annular milling procedure for the fabrication of micro-columns. Thecolumn geometry is determined by the initial height, H0, and the di-ameter at the top, Dtop, and at the foot, Dfoot, of the column. Theaverage diameter, D, and the taper angle are calculated assuming thatthe diameter increases linearly from the top to the foot.

A set of micro-columns was fabricated according to the procedure described pre-viously. Due to the need to create single crystalline volumes, the average diametersof the columns were in the rage between 1.5 µm to 2.0 µm and the initial height wasin the range was in the range between 5.5 µm and 7.0 µm; the aspect ratio is held inthe range between 1 : 2 and 1 : 4. For further investigations into the size dependenceof the mechanical behavior, two columns with nominal diameters of 1 µm and 3 µm,aspect ratios of 1 : 3 (diameter : height) and similar orientations were fabricated.As shown later in this chapter, the specimen size does not influence the mechanicalbehavior for the considered range of specimen dimensions. The micro-columns havethe shape of a tapered cylinder such that the geometry of each column is given bythe initial height, H0, and the initial diameters on the top, Dtop, and at the foot,Dfoot.

The micro-columns were compressed making use of the same nanoindenter asused for the bending experiments. The indenter was equipped with a flat punch tip

41

5 Determination of the Plastic Deformation Properties of the Grains

with a diameter of 15 µm in order to create a uniform displacement at the head of thecolumn. The outer diameter for the milling procedure was chosen as 25 µm such thatthe indenter tip does not make contact with the material surrounding the column.The experiments were carried out using the displacement control with a constantnominal strain rate, ϵ, of 0.0005 s−1. This strain rate was achieved by adjusting thedisplacement rate, h, for each column of the initial height, H0, according to:

h = ϵ H0. (5.1)

In addition to the load-displacement data the elastic stiffness of the column andits support was measured by making use of the dynamic indentation technique(continuous stiffness measurement, CSM). For the CSM measurement a displace-ment amplitude of 2 nm and a frequency of 45Hz have been used. An example fora column before and after compression is shown in figure 5.2. The correspondingload-displacement data is shown in figure 5.3. The geometry of the tested specimensis given in table 5.1.

(a) (b) (c)

Figure 5.2 Example for a micro-column (a) before and (b) after the compressionexperiment. (c) Inverse pole figure describing the orientation of thecompression axis in relation to the grain.

Data Analysis

The steps on the tested column in figure 5.2(b) indicate that plastic deformationtook place along distinct slip planes. Moreover, the plastic deformation is restrictedto the upper part of the column where the stresses are expected to be maximum dueto the tapered shape of the column. The column shown in figure 5.2(b) has beendeformed to strains larger than 30%. The flattened region next to the circular initialcontact area at the top of the column is formed when the indenter makes contact withthe part of the column below the slip region. As expected from other microscopic

42


Figure 5.3 Load-displacement response of the micro-column shown in figure 5.2.

single crystal deformation experiments [16, 21] a mostly linear increase of the loadis followed by discontinuous plastic deformation with displacement bursts.

The discontinuous plastic deformation is attributed to dislocation movement ondistinct slip systems. Since the aluminum alloy has a FCC crystal structure thesingle crystals are expected to deform on 1 1 1 crystallographic planes and in[1 1 0] crystallographic directions. Including all permutations, 12 independent slipsystems are available for the deformation. In order to determine the number of slipsystems that are active during a compression experiment the Schmid factors arecalculated. The Schmid factor, m, relates the normal stress ς to the resolved shearstress τ , i. e. the shear stress acting on a slip system:

τ = m · ς. (5.2)

Knowing the Schmid factors for a single crystal is particularly useful since theyallow to distinguish between experiments where initially only one slip system isactive and experiments where slip is initiated on multiple slip systems. For themeasured orientation of the column along the column axis, as given by the inversepole figure in figure 5.2(c), a maximum Schmid factor of 0.48 is achieved. The nextlower Schmid factor is 0.40, suggesting that only one slip system is active duringthe beginning of the plastic deformation. However, at later stages of the plasticdeformation more slip systems are activated due to changes in the geometry andthe continuously changing crystallographic orientation [66].

Due to imperfections in the contact between indenter and column stemmingfrom surface roughness due to redeposition during the milling process the point offull contact is not consistent with the point of initial contact. In order to assessthe column deformation the following procedure is applied. First, as shown infigure 5.4(a) the local slope of the loading curve, which will be called loading stiffness,

43


Specimen Dtop, µm Dfoot, µm H0, µm

CC-1 1.76 2.46 6.52CC-2 1.70 2.30 7.98CC-3 1.70 2.23 5.46CC-4 1.61 2.19 6.28CC-5 1.59 2.21 6.63CC-6 1.67 2.08 5.65CC-7 1.63 2.18 6.10CC-8 1.8 2.16 6.26CC-9 1.82 2.21 6.77

CC-10 1.76 2.13 6.4CC-11 1.91 2.15 6.55CC-12 1.92 2.13 6.87CC-13 1.91 2.27 6.32CC-14 1.44 1.62 6.57CC-15 0.96 1.21 3.45CC-16 2.73 3.35 8.90CC-17 0.81 1.04 3.64

Table 5.1 Geometries for the tested micro-columns. The aspect ratio is calculatedusing the average diameter.

is determined by using a linear fit over a range of ± 3 nm† around each point of theloading curve. The maximum of the slope, Sfc, is associated with the point offull contact, where effects of the surface roughness on the top of the column arenegligible. As illustrated in figure 5.4(a), the yield point, (hY, PY), is defined hereas the point where the slope of the loading curve becomes less than a thresholdvalue of 20% of Sfc.

The elastic analysis of the mechanical behavior of the columns is circumventedby the uneven foot of the column due to the anisotropic milling rates of the indi-vidual grains adjacent to the column. However, since the column diameter at thefoot is larger than at the top, the highest stresses are expected at the top of thecolumn. Therefore, the plastic deformation is restricted only to the upper part ofthe column, while the foot of the column is assumed to be subject to purely elasticdeformation [28]. Since only the displacement associated with the plastic deforma-tion is of interest for the determination of the parameters of the crystal plasticitymodel, an explicit analysis of the elastic material behavior is not needed. In orderto obtain the displacement associated with the purely plastic deformation, hpl, the

†±20 measuring points since the data acquisition rate is 10Hz

44

5.2 Crystal Plasticity Model for Finite Element Simulations

experimental data must be corrected for the displacement associated with the elasticdeformation, hcorel , as illustrated in figure 5.4(b):

hcorel (P ) =P

Sfc. (5.3)

Assuming that the plastic deformation starts at the yield point, (hY, PY), devia-tions from the linear loading before the yield point require an additional correction,hcorconst. This correction is found by extrapolating linear loading behavior to zeroload, starting at the yield point:

P = 0 = Sfc(hcorconst − hY) + PY, (5.4)

therefore:

hcorconst = hY − PY

Sfc. (5.5)

The displacement associated with plastic deformation, hpl(P ), is then calculated as:

hpl(P ) = h(P )− hcorel (P )− hcorconst (5.6)

⇒ hpl(P ) = h(P )− P

Sfc−

(hY − PY

Sfc

). (5.7)

where h is the total displacement of the indenter.In order to compare specimens with differing geometries the engineering stress,

σ, and the engineering plastic strain, ε, are defined as:

σ =P

A=

Pπ4 D

2(5.8)

and

ε =hplH0

, (5.9)

where A is the area at the center of the column and D is the average diameter,which is calculated using the diameter of the column at the top, Dtop, and at itsfoot, Dfoot:

D =Dtop +Dfoot

2. (5.10)

This normalization can be applied to the experimental data as well as to the simu-lations.

5.2 Crystal Plasticity Model for Finite ElementSimulations

Both the elastic and the plastic deformation of the grains are anisotropic due to thecrystal structure. Using finite element simulations this deformation is reproduced

45


(a)

(b)

Figure 5.4 (a) Definition of the yield point based on the loading stiffness of amicro-column. (b) Correction to the experimental compression dataof a micro-column to obtain the displacement associated with plasticdeformation starting at the yield point.

46

5.2 Crystal Plasticity Model for Finite Element Simulations

with a crystal plasticity constitutive model. In the following the constitutive equa-tions for the model used within this work are given. The origin of this model is acrystal plasticity user material subroutine (UMAT) developed at Harvard Univer-sity [67], which bases on the framework of Peirce et al [68] and Asaro [69, 70]. TheUMAT used in this work has been modified by Graff [71].

The deformation gradient, F, of a crystal lattice can be multiplicatively decom-posed into a part referring to the stretching and the rotation of the lattice, F∗, andthe deformation gradient associated with the plastic deformation, Fp:

F = F∗Fp (5.11)

The velocity gradient therefore can be decomposed in a part referring to the stretch-ing and the rotation of the lattice, L∗, and a plastic part, Lp:

L = FF−1

= L∗ + Lp

= (D∗ +Ω∗) + (Dp +Ωp)

(5.12)

where (D∗,Ω∗) are the lattice parts of the rate of the stretching tensor and thespin tensor, respectively. Accordingly, (Dp,Ωp) are the plastic parts of the rateof the stretching tensor and the rate of the spin tensor. Taking into account thatthe plastic deformation is constrained to specific slip directions, mα, on particularlattice planes described by their normals nα, the plastic velocity gradient, Lp, forthe deformation is expressed by the sum of the slip rates on all slip systems:

Lp =∑α

γαmα ·nα, (5.13)

where γα describes the slip rate on the slip system α. nα and mα are given in thecurrent configuration.

The constitutive law for elastic deformation in a rotating coordinate system canbe written as:

σσσ∗+ σσσ(I : D∗) = C : D∗ (5.14)

with C being the tensor of elastic constants, σσσ the Cauchy stress, andσσσ∗the Jau-

mann rate of the Cauchy stress formed on the axes that spin with the crystal lattice.This rate is related to the corotational stress rate on axes rotating with the material,σσσ, by:

σσσ∗=

σσσ + (Ω−Ω∗) ·σσσ − σσσ · (Ω−Ω∗)

withσσσ = σσσ −Ω ·σσσ + σσσ ·Ω

(5.15)

The plastic deformation on distinct slip systems is assumed as a viscoplasticpower law. The slip rate on each slip system α depends on the resolved shear stressτα:

γα

γα0=

∣∣∣∣ταταY∣∣∣∣n · sign

(τα

ταY

). (5.16)

47


Here ταY characterizes the resistance to slip on the slip system α, i. e. the criticalresolved shear stress (CRSS). γα0 is a reference shear rate and n is the rate sensitivityexponent. The choice of the visco-plastic model for the slip systems allows for anumerically stable way of distinguishing between active and inactive slip systems.

The evolution of the resistance to slip during the deformation process is describedby:

ταY =∑β

hαβ(γ) γβ, (5.17)

with the hardening moduli hαβ , that describe the self hardening (α = β) on a slipsystem and the latent hardening (α = β) due to plastic deformation on other slipsystems. γ is the cumulative shear strain on all slip systems and can be calculatedfrom:

γ =∑α

t∫0

|γα| dt∗. (5.18)

The interaction between slip systems is described by using the interaction coef-ficients, qαβ , and a hardening law, h(γ):

hαβ(γ) = qαβ h(γ) (5.19)

The hardening law in this model is described by an exponential saturation law:

h(γ) = h0

(1− τ0

τ∞

)exp

(−h0 γ

τ∞

), (5.20)

with the initial hardening modulus, h0, and the saturation hardening shear stress,τ∞.

The resistance to slip at each instant t is then given by:

ταY = τ0 +

t∫0

h(γ)

∑β

qαβ γβ

dt∗. (5.21)

The integration constant, τ0, is a measure for the initial resistance against slip foreach slip system, i. e. the initial critical resolved shear stress.

Model Set-up and Handling of the Orientation

The material model has been implemented in a UMAT in the framework of the finiteelement software ABAQUS. The grains of the model material show a face centeredcubic (FCC) crystallographic structure. The elastic constants for AA2198 have beenestimated as given by equation 4.19.

For FCC metals the plastic deformation is constrained to slip on 1 1 1 planesin [1 1 0] directions. In order to correctly determine the resolved shear stress on eachslip system, the relation between the crystal orientation and the global orientationmust be established. EBSD measurements provide the orientation in terms of Euler

48

5.3 Experimental Determination of the Crystal Plasticity Parameters

angles. An orthogonal rotation tensor, R can be calculated from these angles. Thisrotation tensor, R, defines the rotation of the crystal coordinate system into theglobal coordinate system, i. e. the components of the crystallographic vector, vcrystal,are expressed in global coordinates by:

vglobal = Rvcrystal. (5.22)

With this relation given, material orientations for the crystals in ABAQUS aredetermined from three points: the origin (0, 0, 0) and the components of the crys-tallographic vectors (1, 0, 0) and (0, 1, 0) in the global coordinate system. The pa-rameters for the constitutive model need to be defined only once in the ABAQUSinput file and the material orientations are defined for each grain in the definitionof a material section.

The material is assumed to be rate independent. Therefore, the visco-plasticityexponent, n, in equation (5.16) should have a large value. For reasons of numericalstability of the simulations a maximum value of 20 is used in this work. The referencestrain rate has been fixed to 0.001 s−1. These parameters for the visco-plasticity havebeen used throughout the present work.

Finally, the hardening parameters need to be determined such that the load-displacement data of columns is correctly reproduced by the FE calculations. Inorder to reduce the number of fitting parameters, three cases can be distinguished:self hardening for one slip system, interaction of active slip systems on one slipplane (coplanar slip) and interaction of slip systems acting on different slip planes(non-coplanar slip). Thus, the interaction coefficients can be reduced to:

qαβ =

1 if α = β

qcp for coplanar slip

qncp for non-coplanar slip.

(5.23)

5.3 Experimental Determination of the CrystalPlasticity Parameters

Displacement bursts during the plastic deformation have been found in compressionexperiments on micro-columns of various metals (e. g. [16, 21, 72]). This discontin-uous plastic deformation is attributed to the discrete nature of dislocation activity.The crystal plasticity model used here describes the anisotropic deformation withinstrain levels where differences in the geometry between the experiment and thesimulation are small, even though the model cannot account for discontinuous de-formation. From the shape of the load-displacement curves of several specimensthe plastic deformation shows a tendency to saturate. The exponential saturationhardening law, as presented in section 5.2, provides a good representation for thissaturation within the FE model. The remaining parameters to be determined arethe self hardening parameters and the interaction coefficients. The self-hardeningparameters are the initial flow stress, τ0, the initial hardening modulus, h0, and the

49


saturation hardening stress, τ∞. The interaction coefficients are given as qcp andqncp for coplanar and non-coplanar slip, respectively.

Finite element models for each micro-column were created. These models forthe micro-columns are three dimensional models that include the initial height aswell as the diameters at the foot and on top of the column. An example FE-meshis shown in figure 5.5. The column is smoothly connected to its support whichconsists of a cylindrical disk of 10 µm in height. The orientation of the respectivecolumn was assigned to the complete model; orientations of adjacent grains havebeen neglected. The ABAQUS mesh generator was used to create the finite elementmesh that consists of 8-node brick elements (C3D8). The approximate length of eachelement edge was requested to be smaller than a tenth of the columns diameter onthe top. The mesh in the support coarsens with increasing distance to the column.The bottom surface of the support was subject to fixed boundary conditions. Theindenter was modeled by a prescribed axial displacement on the top surface of thecolumn not allowing for lateral movement of any node of the top surface. Thus, theexperiment was modeled assuming ideal adhesion of the indenter to the top surfaceof the column and no lateral compliance of the indenter. The displacement rate wasset to the average displacement rate during the experiment which is determined bythe maximum displacement at 10% engineering strain, divided by the time neededto achieve this displacement. The investigation of the deformation is limited to thedisplacement where 10% of the engineering strain is achieved. Beyond that pointthe geometry of the deformed column significantly differs from the geometry foundin the simulation due to the discrete deformation. The parameters of the crystalplasticity model are, thus, fitted to this limit of 10% engineering strain.

Figure 5.5 Example FE-mesh for modeling of the deformation of a micro-column.

For the determination of the crystal plasticity constants a column with defor-mation on one distinct slip system was selected. This allows to determine the selfhardening parameters, τ0, τ∞ and h0, of one slip system without influence of otherslip systems. The load-displacement data determined from the simulation was com-pared to the deformation of the corresponding specimen which was determined fromthe analysis procedure given in section 5.1. A set of simulations was carried outto determine optimized parameters that allow for the correct reproduction of theload-displacement curve (see figure 5.6). Within these simulations both interaction

50


τ0,MPa τ∞,MPa h0,MPa qcp,- qncp,-

120.0 137.0 8000.0 2.0 10.0

Table 5.2 Crystal plasticity constants determined form micro-compression experi-ments.

coefficients, qcp and qncp, were set to unity. The optimized parameters are given intable 5.2. Figure 5.7 qualitatively compares the deformation from the experimentand corresponding FE-simulation. The global slip direction of the column in theFE simulation is identical to the slip deformation in the corresponding experiment.Moreover, only the slip system with the highest Schmid factor is active in the sim-ulation of the specimen oriented for single slip, which is shown in figure 5.8. Theplastic deformation in the simulation is restricted to the upper part of the columnconfirming the assumption that the influence of the foot on the plastic deforma-tion is negligible. The secondary slip system remains inactive until later stages ofdeformation.

Figure 5.6 Load-displacement response of the specimen oriented for single slip(CC-1). Open symbols represent the simulation results, solid symbolsgive the experimental data.

After the self hardening parameters were determined, the interaction parameterswere adjusted to reproduce the deformation of columns with more than one initiallyactive slip system. The experimental load-displacement data of a column withtwo active coplanar slip systems was used to quantify the interaction coefficientqcp. Another column with two non-coplanar slip systems provided the experimentaldata to determine the interaction coefficient qncp. The interaction coefficients arealso given in table 5.2. These coefficients differ form the coefficients usually used incrystal plasticity simulations which are 1 for coplanar slip and 1.4 for non-coplanarslip (see [73]). Figure 5.9 depicts the experimental load-displacement data of the

51


Figure 5.7 Comparison of the deformations of a tested specimen (CC-1) and thecorresponding simulation. The simulation depicts the result corre-sponding to 10% nominal strain while the maximum nominal strainin the experiment was 30%. The slip direction in the experiment isidentical to the slip direction in the simulation.

(a) (b)

Figure 5.8 Simulation of a deformed column (CC-1) shown in figure 5.2. Thecolor code represents the shear strain, γ, (a) on the slip system withthe highest Schmid factor, 0.48, and (b) the slip system providing thesecond highest Schmid factor, 0.40.

52


columns used for the parameter optimization together with the load-displacementdata extracted from the respective simulations.

Figure 5.9 Load-displacement response (solid symbols) and FE-prediction (opensymbols) for the columns used for the fitting of the crystal plasticityparameters (CC-1, CC-14 and CC-15).

The material model was verified by applying the parameter set to the FE sim-ulations of the remaining columns. The results for three columns are shown infigure 5.10. The material model predicts the load-displacement response of columnswith various orientations quite well.

In order to provide limits for the application of the crystal plasticity parametersthe average value of the accumulated plastic strain, γ, at an engineering plastic strainof 10% in a column without the support was calculated. For the column orientedfor single slip as given in figure 5.8 this value is on the order of 20%. If duringthe application of the adjusted crystal plasticity model values beyond this limit arefound at material points, the respective material behavior must be considered as anextrapolation of the model.

In the case of columns with the compression axis close to a [1 1 1] crystal orienta-tion the material model overestimates the experimental yield stresses, as can be seenfrom figure 5.11. The reasons for this overestimation are probably related to the mi-croscopic deformation mechanisms that govern the deformation of the micro-column.This deviation might be explained with the potentially unstable deformation dueto compression in connection with the particular symmetry of this orientation butwas not further investigated. However, it is shown in the following chapter that theoverestimation has no influence on the simulation of the deformation of cantilevers.

Size Effects in Micro-Compression Tests

In the case of micro-column compression experiments on single-crystalline mono-lithic specimens, it is widely observed that the yield stresses increase when the

53


Figure 5.10 Examples for the application of the calibrated crystal plasticity modelto columns with various orientations (CC-3, CC-7 and CC-17). Solidsymbols are experimental data, open symbols represent simulationdata.

Figure 5.11 Load-displacement response of specimens with the compression axisclose to a ⟨1 1 1⟩ crystal orientation (solid symbols; CC-9, CC-12 andCC-13). Open symbols represent the simulation results.

54


diameter of the micro-columns is decreased [16, 21], thus, for the present study anestimation of the influence of the size of the specimens on the deformation behavioris necessary.

Columns with nominal diameters of 1 µm, 2 µm and 3 µm and constant aspectratios of 1 : 3 (diameter : height) were fabricated out of grains of similar orientationsand tested with the same method as the columns presented in this chapter. Finiteelement simulations for each of the columns using the fitted parameter set were car-ried out and the results were compared to the experimental load-displacement dataof the compression experiments. The load-displacement data is in good agreementwith the experimental data from the compression experiments of all column sizes,as can be seen from figure 5.12. It can be concluded that crystal plasticity param-eters, as determined from the micro-compression experiments on single crystallinecolumns of the model material, are not influenced by the size of the specimens inthe investigated range between 1 µm and 3 µm. The absence of a size effect is notunexpected. Due to the alloying content and the thermo-mechanical treatment themicrostructural lengthscale controlling the stress-strain behavior is much smallerthan the geometric length scale as given by the diameter of the column. This is incontrast to well annealed single crystals of pure metals in which most size effectsare observed.

Figure 5.12 Load-displacement response of specimens with nominal diameters of1 µm, 2 µm and 3 µm providing similar orientations along the columnaxis (CC-11, CC-15 and CC-16). The aspect ratio is 1 : 3 for eachcolumn. Solid symbols represent the experimental data, open symbolsprovide data extracted from FE-simulations.

55


Effects of the Boundary Conditions on the Simulation ofMicro-compression Experiments

The boundary conditions can have a significant impact on the load-displacementdata of micro-compression tests. For the simulations it was assumed that the surfaceat the foot of the column is smooth and the orientation of the surrounding materialwas neglected. Due to the tapered geometry of the column these regions weresubject to elastic deformation only. Since the analysis of the micro-compressiondata excludes the elastic deformation, the assumed simplifications have no effect onthe result.

The boundary conditions on top of the column can be important for the plasticdeformation behavior. It is to be distinguished between effects induced by the lateralstiffness of the indenter and the friction between the indenter head and the top ofthe column. The lateral indenter stiffness is on the order of 10000N/m while thelateral bending stiffness of the columns is on the order of 1000N/m. Therefore,an indenter with infinite lateral stiffness was assumed. The friction between theindenter head and the column has not been quantified for the micro-compressiontests presented here. SEM-images of tested columns show that the top surface ofthe columns did not alternate its shape plastically such that ideal adhesion betweenthe indenter tip and the surrounding material is a reasonable approximation.

Summary

• Micro-column compression experiments were used to determine the load-dis-placement response of single crystals subjected to uniaxial stresses.

• A finite element crystal plasticity model was adopted to describe the defor-mation of single grains within a finite element framework.

• A set of crystal plasticity parameters was identified from the compression tests.These parameters are valid up to values of 20% accumulated plastic strain.

• The material model was verified for various orientations with the exceptionof the case where orientation of the compression axis of the column is alignedwith a [1 1 1] crystal orientation. The quality of the prediction for variousorientations is summarized in figure 5.13.

• Influences on the measurement of crystal plasticity constants were discussed.In particular, the results were independent of the size of the columns.

56


Figure 5.13 Anisotropic yield according to various crystal orientations (solid sym-bols). Open symbols represent the simulation results.

57

6 Determination of the GrainBoundary Strength

The determination of the strength of a single grain boundary requires the com-bination of the experimental technique as established in chapter 4 and the finiteelement based crystal plasticity model for the single grains presented in chapter 5.The integration of the crystal plasticity model into the finite element simulations forthe notched cantilever specimens is presented here. Additionally, a cohesive zonemodel is introduced to the finite element framework to investigate the separationproperties of a single grain boundary. Finally, the methodology is applied to twocantilevers that fractured after significant plastic deformation and shortly after theonset of plastic deformation, respectively. The results are discussed in terms of grainboundary strength.

6.1 Transfer of the Crystal Plasticity Model to theCantilever Specimens

Finite element models for each cantilever were built according to the models pre-sented in section 4.6 including the lateral position and the lateral stiffness of theindenter. The isotropic elastic material was replaced by the crystal plasticity modelincluding the parameters determined in chapter 5. The cantilever was thereforepartitioned into four sections as shown in figure 6.1(a). The two sections adjacentto the grain boundary were assigned the material orientations determined by EBSDmeasurements. In order to simplify the simulations, a specific material orientationwas assigned to the sections not adjacent to the notch such that a [1 1 0] crystal-lographic axis was aligned with the cantilever axis. This is reasonable since simu-lations have shown that the crystallographic orientation of these sections has littleeffect on the load-displacement data and that this orientation is frequently foundalong the cantilever axis. The displacement at the grain boundary is prescribedto be continuous without further restrictions to the deformation. The cantileverswere loaded in displacement control using the average displacement rate identifiedfrom the experiments. In addition to the cantilevers with a grain boundary at theposition of the notch, cantilevers with a notch in a single grain were fabricated. Thecorresponding orientations of the cantilever sections are given in figure 6.1(b).

Assessment of the Load-Displacement Response of the Cantilevers

The cantilevers with the notch placed in a single grain allow the assessment of thedeformation of a grain during the bending experiment separately from any influence

58

6.1 Transfer of the Crystal Plasticity Model to the Cantilever Specimens

(a) (b)

Figure 6.1 Orientations within the cantilever model (a) with the notch at the po-sition of a grain boundary and (b) with the notch placed within a singlegrain.

of a grain boundary under the notch. In figure 6.2 the experimentally recordedload-deflection data for these cantilevers is compared to the corresponding data ex-tracted from the respective simulations. Figure 6.3 shows the data for cantileverswith a grain boundary under the notch. Generally, the simulation data was in goodagreement with the experimental data. However, differences between the experi-ments and the simulations exist. Possible reasons are discussed at the end of thissection.

Figure 6.2 Comparison of the experimental load-deflection response of two can-tilevers with the notch placed in a single grain (solid symbols) with thecorresponding simulation data (open symbols).

Local Deformation in the Vicinity of the Notch

The simulations must capture the local deformations of the grains adjacent to thenotch. In order to assess these deformations local measurements of the deformationfield are required.

59

6 Determination of the Grain Boundary Strength

Figure 6.3 Comparison of the experimental load-deflection response of three can-tilevers with a grain boundary under the notch (solid symbols) withthe corresponding simulation data (open symbols).

Making use of an in situ test setup in the SEM, the displacement field aroundthe notch was captured using digital image correlation, as described in section 3.4.In order to enable such measurements a platinum speckle pattern was created onthe side surface of a cantilever in the region under the notch. The deposition of theplatinum was achieved with the electron beam in combination with a gas injectionsystem (GIS) to prevent any additional ion damage. The platinum pattern consistedof a large number of platinum dots such that the mechanical behavior of the can-tilever was not affected by the pattern. The size of the dots was chosen accordingto the expected displacements, the desired magnification and the image resolution(pixel size) available. In order to capture the displacement field in the region mea-suring 8 µm along the cantilever axis and 6 µm along the cantilever height under thenotch, platinum dots with average diameter of 200 nm were deposited. The spacingbetween the platinum dots was on the same order as the diameter

A reference image of the undeformed cantilever was recorded. The cantileverwas then bent to a load of 1.6mN making use of an in situ nanoindenter (Nanome-chanics Inc.) equipped with a cube corner tip. The force on the cantilever was heldconstant while high resolution images of the deformed cantilever were recorded. Thedisplacement field at the notch was determined by applying DIC to the referenceimage and the deformed image. The size of the subsets of pixels was on the orderof the size of the platinum dots. Figure 6.4 shows the displacement field for thecantilever with a grain boundary under the notch. The experimentally determineddisplacement field and the displacement field extracted from the corresponding sim-ulation at the same load were generally in good agreement. It can be concluded thatnot only the load-displacement curves but also the local deformations are capturedby the simulations. The scatter resulting from the DIC measurement was on theorder of ±5 nm while the maximum displacement was on the order of 50 nm. Hence,

60

6.1 Transfer of the Crystal Plasticity Model to the Cantilever Specimens

the determination of the local strain field from the DIC data leads to a significanterror. Inaccuracies in the displacement field under the notch may result from thescatter in the displacement measurement. Possible inaccuracies related to the finiteelement modeling are discussed at the end of this chapter.

(a) (b)

Figure 6.4 Displacement field at the notch ground of a cantilever measured bydigital image correlation (a) and extracted from the corresponding FEMsimulation at the same load (b).

Deviations between Simulations and Experimental Data

The load-displacement data determined by the simulation is highly sensitive tothe measured values of the characteristic dimensions. Figure 6.5(a) illustrates theeffect of the geometry of the cantilever on the load-displacement data. The greentriangles represent the load-displacement data extracted from a simulation of acantilever with the nominal dimensions height : width : length = 8.0 µm : 5.0 µm :20.0 µm. The black and red lines are the upper and lower limits calculated byvarying the cantilever dimensions, respectively. The precision of the measurementof the cantilever dimensions from SEM images was better than 0.2 µm. Hence, theupper limit was achieved by adding 0.2 µm to the height and the width and byreducing the length by 0.2 µm. Accordingly, the lower limit was calculated with thewidth and the height reduced by 0.2 µm while the length was increased by 0.2 µm.These limits represent the worst case for deviations in the load-displacement datacaused by the measurement of the geometry of the cantilevers.

While the global load-displacement response provides an overview of the defor-mation process the, local stress field at the grain boundary determines the fractureprocess. The evolution of the normal stress in the direction of the cantilever axis atthe center of the notch ground was evaluated for the upper and the lower limitingcases in order to provide an estimation of the effect of the geometry on the local

61


stresses. In figure 6.5(b) this stress is given for the limiting curves. The differencesin the geometry only slightly alter the stress at the notch ground.

(a)

(b)

Figure 6.5 (a) Load-deflection response of a cantilever with the dimensions height :width : length = 8.0 µm : 5.0 µm : 20.0 µm together with the upper andlower limits due to deviations in the geometry. (b) Stress at the centerof the notch ground for the upper and lower limits.

Deviations of the hardening parameters of the crystal plasticity model also in-fluence the load-displacement response of a cantilever and the stresses at the grainboundary. In particular, the saturation stress, τ∞, and the initial hardening mod-ulus, h0, (see section 5.2) determine the stress at a material point after the onsetof plastic deformation. Starting with the cantilever geometry associated with thenominal dimensions, the saturation stress was varied by ±10MPa and the initial

62

6.2 Cohesive Zone Model of the Grain Boundary

hardening modulus was varied by ±4000MPa. Figure 6.6(a) illustrates the effectof the variation of the crystal plasticity parameters on the load-displacement curve.The variation of the saturation stress causes an increasing difference of the loadwith increasing deflection of the cantilever. The variation of the hardening modulushas only little influence on the data during the initial stages of plastic deformationand the differences even decrease at larger deflections.

The evolution of the normal stress along the cantilever axis at the center of thenotch ground was determined from the simulations to estimate the effect of thevariation of the crystal plasticity parameters on the fracture process 6.6(b). Inparticular, the saturation stress (solid lines) has a strong influence on the stress atthe grain boundary. In contrast, the influence of the initial hardening modulus ismoderate (dashed lines).

It is important to note that the crystal plasticity constants were determined toa maximum accumulated plastic strain of 20%, as shown in chapter 5. Therefore, ifthe accumulated plastic strain in the cantilever exceeds 20% the material responseat the respective material points must be considered as an extrapolation of the vali-dated material model. The plastic strain within must be assessed for each cantileverexplicitly since the geometry and crystal orientation can cause significant differencesof the maximum strain between specimens.

In conclusion, deviations of the load-displacement response between the exper-imental data and the simulations of cantilevers can be caused by uncertainties inthe measurement of the geometry and deviations in the parameters of the crystalplasticity model. The stress in the vicinity of the grain boundary is less sensitive tothe geometry but strongly depend on the saturation stress provided by the crystalplasticity model. Hence, a precise determination of the crystal plasticity parametersis required.


Figure 6.7 shows the distribution of the normal stress at the grain boundary asprovided by the respective finite element simulation. A fracture criterion that isindependent of the stress distribution at the grain boundary is required to analyzethe fracture of a grain boundary. The cohesive zone modeling provides such adescription within the finite element framework [74]. In this section the cohesivezone approach applied within the present work is described.

Description of the Applied Cohesive Zone Model

Since the thickness of a grain boundary is negligibly small, a constitutive law whichrelates the traction directly to the separation of the crack surfaces was chosen. Sucha constitutive behavior is also called a traction-separation law which has three keyaspects:

• the elastic part without damage

63


(a)

(b)

Figure 6.6 (a) Load-deflection response of a cantilever with the dimensions height :width : length = 8.0 µm : 5.0 µm : 20.0 µm. The red lines represent thedeviations due to the variations of the initial hardening modulus of±4000MPa. The black lines represent the deviations when the satu-ration stress is varied by ±10MPa. (b) The evolution of the normalstress at the center of the notch ground.

64


Figure 6.7 Normal stress along the cantilever axis at the grain boundary at thefracture load predicted by a finite element simulation.

• the initiation criterion that defines when damage starts

• the evolution of damage until complete failure

Figure 6.8 gives an illustration of the constitutive behavior of the bilinear traction-separation law used within this work.

Figure 6.8 Representation of the traction-separation law applied within thepresent work.

Both the tractions and separations have three components: one normal com-ponent perpendicular to the surface and two shear components within the surfacefollowing an orthonormal coordinate system. For the present case only the traction-separation behavior normal to the grain boundary plane is considered since no shear

65


deformation or failure at the grain boundary was observed in the experiments pre-sented here. Within the finite element code ABAQUS the elastic part is assumedto be linear elastic giving:

t = K · εδ = K · δe, (6.1)

with t denominating the traction (i. e. stress) normal to the surface, δ being theseparation of the crack surfaces and the elastic interface stiffness in absence ofdamage, K. The initial thickness of the cohesive layer is denoted by e and thestrain in the cohesive layer is denoted by εδ. Since the cohesive zone represents atwo dimensional surface with an infinitesimal thickness, e serves as a parameter forthe calculation of the interfacial stiffness and is set to unity.

Damage is initiated when the stress normal to the surface is equal to the initiationtraction, ti. The initiation of damage by shear deformation is prohibited by themodel used in this work since normal separation of the grain boundaries has beenobserved to be the dominant failure mode. Potential consequences of the limitationto normal damage initiation are briefly discussed at the end of this chapter. Thedamage initiation criterion is expressed in terms of the normal stress, t:

t

ti= 1. (6.2)

It is important to note that only tensile traction generates damage while compressivestresses do not lead to the initiation of damage.

Finally, the evolution of damage needs to be specified. A damage variable, D,represents the damage in the material where D = 0 is undamaged material andD = 1 represents complete failure. The normal cohesive stress would be, t, if nodamage was present in the cohesive zone. This stress is reduced factor of (1 −D)to calculate the actual cohesive stress after damage initiation, t:

t = (1−D) · t. (6.3)

Starting at the initiation point, (δi, ti), the traction continuously decreases until fullfracture is achieved at D = 1, thus t = 0. Thereby, the maximum separation (i. e.the separation at complete failure) δf , is attained which is related to the fractureenergy, Gf . For the bilinear traction-separation law the damage variable is given as:

D =δf(δ − δi)

δ(δf − δi)(6.4)

where δf indicates the separation at complete failure. Following this approach thefracture energy, Gf , is expressed as the area under the traction-separation curve:

Gf =1

2δf ti. (6.5)

Once damage is initiated the material response for unloading is linear elastic with areduced slope, K(1−D), compared to the initial slope during loading, K. Finally,

66


the reloading of the material is subject to the linear elastic behavior with the samereduced stiffness. It is important to note that the definition of damage for thecohesive zone does not correspond to the definition of damage of the cantilever asgiven in chapter 4. There, damage is described as the global loss of elastic stiffness ofa cantilever due to a crack at the grain boundary. Damage in terms of the cohesivezone defines the local loss of load carrying ability at a cohesive element. Only if alarge number of cohesive elements are damaged the elastic stiffness of the cantileveris reduced significantly.

Application within the Finite Element Framework

For each cantilever one cohesive layer was introduced at the position of the grainboundary. The cantilever model was separated into three parts: the cohesive zonerepresenting the grain boundary, the supported part and the free part (see fig-ure 6.9). These parts were tied together to ensure continuous displacement over thegrain boundary. The meshes of the supported part and the free part were createdanalogously to the mesh used in section 6.1. The mesh for the cohesive zone was cre-ated as an irregular mesh in order to prevent any influences of the mesh structure onthe failure. The elements were three dimensional 8-node elements (ABAQUS spec-ification: COH3D8). The average edge length of the cohesive elements was half theedge length of the continuum elements at the notch. From the traction-separationlaw described previously, three parameters needed to be determined: the initial elas-tic constant K, the fracture energy to describe the damage propagation, Gf , andthe traction at the point of damage initiation, ti. It is important to note that thefracture energy is related to the separation at complete fracture by equation (6.5)such that these parameters are not independent.

Figure 6.9 Integration of the cohesive zone into the cantilever.

The purpose of the cohesive layer is to provide a fracture criterion within thesimulation while the deformation properties are completely captured by the mate-rial surrounding the grains. Therefore, the elastic interface stiffness, K, is chosensuch that the cohesive layer does not influence the deformation behavior of the can-

67


tilever. A value of ten times the isotropic elastic modulus of the material ensuresa sufficiently high normal stiffness of the cohesive layer. Additionally, the cohesiveshear stiffness must be defined such that no shear deformations occur in the cohesivelayer. The elastic shear deformations in both shear directions of the cohesive layerin absence of damage are described analogously to equation 6.1.∗ Therefore, thecorresponding stiffness constant, KS, applies for both shear directions and is set toa value of ten times the isotropic shear modulus.

In chapter 4 it has been shown that the grain boundaries fail without a significantdecrease of the elastic stiffness of the cantilever prior to complete fracture. Thisimplies a small fracture energy for the grain boundary, i. e. grain boundary failureis brittle. The determination of the fracture energy in terms of the cohesive zoneelements requires crack propagation measurements. Since this was not possible withthe present test setup a low value for the fracture energy has been selected. Forpure aluminum bi-crystals subject to liquid metal embrittlement crack extensionforces on in the range between 0.3 J/m2 and 3.0 J/m2 were found depending on themisorientation of the grain boundary [75]. Here a value of 5.0 J/m2 was used inorder to ensure numerical stability of the simulations. The simulation was stoppedwhen the time increment of the quasi static simulation achieved its minimum value of10−5 s. This was defined as the point of final, complete fracture for the correspondinginitiation stress since the quasi static simulations cannot account for the unstablecrack propagation.

6.3 Determination of the Initiation Stress for twoRepresentative Cantilevers

From the fracture experiments three representative cases can be identified: (a)grain boundaries that fracture right after the onset of plastic deformation, (b) grainboundaries where the fracture is initiated after extensive plastic deformation and(c) grain boundaries that do not fracture. In order to assess the capabilities of themethod developed within this work it was applied to two cantilevers which representcases (a) and (b) and which will be called cantilever A and cantilever B, respectively.The post mortem images of these specimens are shown in figure 6.10. It is worthnoting that cantilever A seems to be more strongly deformed than cantilever B.This is basically related to the larger crack length of cantilever A and not to plasticdeformation, as can be seen from the load-displacement data given in figures 6.11and 6.12. The geometry of the cantilevers is given in table 6.1.

The orientation of the grains under the notch of the cantilevers was measuredusing EBSD prior to the fabrication of the specimens. The Cartesian coordinatesystem of the EBSD system with reference to the cantilever can be described asfollows: the y-axis is parallel to the cantilever axis and the z-axis perpendicular tothe top surface of the cantilever (i. g. parallel to the indenter axis). The Eulerianangles in Bunge notation [76] are given in table 6.2 for the respective grains together

∗The traction-separation behavior is uncoupled.

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6.3 Determination of the Initiation Stress for two Representative Cantilevers

(a) cantilever A (b) cantilever B

Figure 6.10 Cantilevers showing fracture (a) directly after the onset of plasticdeformation and (b) after pronounced plastic deformation.

cantilever H, µm L, µm B, µm d, µm a, µm

A 8.70 21.64 6.13 3.13 1.02B 9.43 26.70 7.02 4.55 1.13

Table 6.1 Geometry of the cantilevers A and B. H is the cantilever height, L is thecantilever length, B is the cantilever with, d is the position of the notchcenter and a is the depth of the notch.

with the misorientation angle, θ, referring to a plane grain boundary with its normalparallel to the cantilever axis. Both grain boundaries are high angle grain boundarieswith similar misorientation angles on the order of 50. It is important to note thatthe misorientation angle is used for a coarse characterization of a perfectly planegrain boundary while the inner structure of the grain boundary is not represented,e. g. local deviations of the grain boundary orientation and particles on the grainboundary plane.

The load-displacement data for cantilever A is given in figure 6.11. The grainboundary of cantilever A fractured at a deflection of (1000± 50) nm. The de-flection at the point of fracture was approximated with an initiation stress, ti,of (620± 10)MPa assuming a constant fracture energy of 5 J/m2. Cantilever Bshowed more plastic deformation prior to fracture at a deflection of (1800± 50) nm(see figure 6.12). The deflection at the point of fracture was approximated with aninitiation stress, ti, of (600± 10)MPa when the fracture energy was held constantat 5 J/m2. Hence, both grain boundaries fractured at similar initiation stresseswhile the cantilever deformation prior to fracture was significantly different. This isconsistent with the fact that both grain boundaries showed similar misorientationangles.

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cantilever grain ϕ1 Φ ϕ2 θ

Asupport 148.9 62.1 46.2

47.5free 152.4 112.6 55.9

Bsupport 209.4 108.3 218.3

51.5free 87.0 92.2 0.3

Table 6.2 Orientations of the grains under the notch for cantilevers A and B interms of Eulerian angles (ϕ1,Φ, ϕ2) in the Bunge notation. The label“support” denotes the orientations of the grains close to the supportedend, the label “free” denotes the orientations of the grains close to thefree end of the cantilever. θ is the misorientation angle of the grainboundary.

Figure 6.11 Load-displacement data of cantilever A. The cantilever failed at1000± 50 nm. The initiation stress, ti has been adjusted to matchthe displacement at failure while the fracture energy was kept con-stant at 5 J/m2.

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Figure 6.12 Load-displacement data of cantilever B. The cantilever failed at1800± 50 nm. The initiation stress, ti, has been adjusted to match thedisplacement at failure while the fracture energy was kept constant at5 J/m2.

It is important to note that the simulations did not account for the final frac-ture process; the rapid failure of the grain boundary negates an analysis of crackpropagation. Instead the time increment of the quasi static analysis was reduceddown to a minimum value of 10−5 s prior to complete fracture. This last point ofthe simulation is defined as point of fracture for a simulation within this work.

In order to investigate the influence of the fracture energy on the results, thefracture energy was varied by ±1 J/m2 for both cantilevers while the initiation stresswas kept constant at 620MPa and 600MPa, respectively. For both cantilevers anincrease in fracture energy shifted the final fracture in the simulations to largervalues of cantilever deflection. Accordingly, the lower fracture energies shifted thefinal fracture to lower cantilever deflections. A variation of the fracture energy of20%, as given in figure 6.13, results in a variation of the deflection at fracture on theorder of 10%. The cohesive model is more sensitive to a variation of the initiationstress as can be seen from the fact that a variation of the deflection at fracture onthe order of 10% is caused by a variation of the initiation stress by approximately2%, as given in figures 6.11 and 6.12.

One possible reason for the significantly different deformation behavior of thecantilevers could be the cantilever geometry. The position of the notch, d, relative tothe cantilever length, L, was d

L = 0.14 for cantilever A and dL = 0.17 for cantilever B.

The difference between these relative positions is not expected to influence thefracture behavior significantly. In turn, the analysis of the fracture stress by themethod developed within this work includes effects of the cantilever geometry, whichis a major advantage.

Another possible reason for the differences in the deformation behavior betweencantilever A and B might be related to different stress distributions at the grain

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(a) cantilever A

(b) cantilever B

Figure 6.13 Load-displacement data of (a) cantilever A and (b) cantilever B for avariation of the fracture energy, Gf , between 4 J/m2 and 6 J/m2. Theinitiation stress was 620MPa for cantilever A and 600MPa for can-tilever B. In figure (b) the simulation for a fracture energy of 4 J/m2

stopped before the increment was broken down for unknown reasons.

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boundary. In order to investigate possible influences of the stress distribution, theinitiation criterion and the damage in the cohesive zone were considered togetherwith the distribution of the normal stress along the cantilever axis in the grain closeto the supported end of the cantilever. An initiation stress of 620MPa and a fractureenergy of 5 J/m2 were chosen for the following simulations of cantilever A. Damagewas initiated at a deflection of 750 nm. The value of the damage initiation crite-rion for the cohesive elements along the grain boundary is shown in figure 6.14(a).Damage was initiated at the notch ground when the initiation criterion attained avalue of 1. This is consistent with the stress distribution in terms of normal stressalong the cantilever axis at the grain boundary as depicted in figure 6.14(b). Thestress distribution was not symmetric with respect to the center of the cantilever.Probably, this is caused by the anisotropic deformation of the grains and that thefact that the indenter was not positioned at the center of the cantilever axis. Thisunderscores the importance of a three-dimensional simulation with an accurate rep-resentation of the position of the indenter and the use of an anisotropic materialmodel. After damage was initiated a damage zone was formed around the locus ofinitiation at the notch ground, as can be seen for the last increment of the simulationshown in figure 6.14(c), which provides the final state prior to complete fracture.In this case damage approaches the maximum value of 1. The stress distribution atthe grain boundary at the moment of fracture is shown in figure 6.14(d).

For the following simulations of cantilever B an initiation stress of 600MPa anda fracture energy of 5 J/m2 were chosen. Damage in cantilever B was initiated belowthe notch ground, as can be seen in figure 6.15(a). The initiation of damage startedat a cantilever deflection of 1060 nm. The stress distribution of normal stressesalong the cantilever axis at damage initiation is shown in figure 6.15(b), providingits maximum below the notch ground. Moreover, a local maximum was found evenfurther below the center of the notch ground. Such a stress distribution is likely tobe caused by the anisotropic deformation of the grains. At the last increment of thesimulation, a damage zone was found around the locus of primary damage initiation(see figure 6.15(c)). The stress distribution at the grain boundary within the lastincrement of the simulation is given in figure 6.15(d).

It is worth noting that the normal stress in the crystal sections close to thedamaged zone was up to 10% higher than the normal stress of the cohesive zone.This could be caused by inaccuracies in the formulation of the cohesive elementsin combination with the visco-plastic material used for the crystal plasticity formu-lation. Moreover, it is important to note that the maximum accumulated plasticstrain in the cantilever was locally on the order of 30% for both cantilevers A andB. Accumulated plastic strains beyond 20% result in an extrapolation of the of thematerial response beyond the limits of the determination of the crystal plasticityparameters given in section 5.3. Figure 6.16 shows that the region where the plasticparameters were extrapolated was located at the notch ground. The local stressdistribution can be affected by this model depending on the accuracy of the extrap-olation. However, the extrapolated material behavior only affects small regions suchthat no significant influence on the global load-displacement behavior is expected.

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(a) (b)

(c) (d)

Figure 6.14 Cantilever A: (a) Initiation criterion and (b) stress distribution in thematerial adjacent to the grain boundary at the moment of initiation.(c) Damage and (d) normal stress distribution at the grain boundaryfor the last increment of the simulation.

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(a) (b)

(c) (d)

Figure 6.15 Cantilever B: (a) Initiation criterion and (b) stress distribution atthe grain boundary at the moment of initiation. (c) Damage and (d)normal stress distribution at the grain boundary for the last incrementof the simulation.

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(a) cantilever A (b) cantilever B

Figure 6.16 Regions of the material in the grain close to the supported end wherethe plastic deformation is extrapolated (red domains, γ > 20%) forcantilevers A and B.

This analysis of the stress distribution at the grain boundary depends on thecrystal plasticity model used for the simulations and the related boundary conditionsat the grain boundary. For the present analysis a continuous displacement overthe grain boundary, in absence of complete damage, has been assumed. Otherboundary conditions at the grain boundary can significantly influence the stressfields, e. g. the zero shear condition perpendicular to the interface which makes thegrain boundary a perfect obstacle to dislocations [77]. Furthermore, possible effectsof strain gradients on the stress distribution are neglected within the present model.These effects are subject to current research activities [78, 35]. Moreover, particlesand or precipitates on the grain boundary alter the stress field in the vicinity ofthe grain boundary locally. Finally, the actual structure of the grain boundary maydeviate from the assumption of a perfectly plane boundary.

In conclusion, the model used in the present analysis sufficiently approximates theplastic deformation of the grains in the present model, although, it is not expectedto provide an exact representation of the stress field at the grain boundary. Theinitiation stress in terms of the cohesive zone model is, thus, related to the innerstructure of the grain boundary and possible effects of the boundary conditions andstrain gradients.

Summary

• The adjusted model of crystal plasticity can be successfully transfered frommicro-columns to micro-cantilever specimens.

• The cantilever model has been extended by a cohesive zone for investigationsinto fracture parameters.

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• The method presented here for grain boundary fracture analysis of single brit-tle grain boundaries in ductile metals has been applied to two cantilever spec-imens with significantly different deformation characteristics but similar mis-orientations of the grain boundary of interest. Despite the differences in thedeformation behavior, the method identified similar damage initiation stressesfor both grain boundaries.

• The cohesive zone model for brittle fracture is more sensitive to the initiationstress than to the fracture energy. Therefore, the initiation stress was adjustedfor the brittle fracture of the grain boundary while the fracture energy washeld constant at 5 J/m2.

• The crystal plasticity model approximates the plastic deformation of the grainswith sufficient accuracy to determine the initiation stress in terms of the cohe-sive zone model. This initiation stress is related to the inner structure of thegrain boundary and possible effects of the boundary conditions at the grainboundary and strain gradient effects.

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7 Conclusions

Within the present work a method for the separation of single grain boundaries inplastically deforming metals has been presented. Notched micro-cantilever speci-mens were fabricated from the aluminum-lithium alloy 2198 using focused ion beammilling with a gallium ion source. The specimens were fabricated with a U-notchplaced at the boundary of interest close to the supported end of the cantilever. TheFIB fabrication lead to the embrittlement of the grain boundaries which helped toincrease the number of successfully tested specimens; a great benefit for the devel-opment of the method. Using a nanoindenter the cantilevers were bent to fracture.The cantilevers were deformed plastically and no significant decrease of the elasticstiffness was found prior to fracture. It can be concluded that the fracture of thegrain boundaries is brittle while all permanent deformation of the cantilevers canbe attributed to the plastic deformation of the grains. Making use of a finite ele-ment model including an isotropic elastic constitutive law, specimens with varyinggeometries were compared to each other. In the analysis of the specimens it wasfound that the actual lateral stiffness of the indenter must be included in the finiteelement simulations to provide accurate results.

In order to capture the anisotropic plastic deformation of the grains a crystalplasticity finite element model was applied. It has been shown that in technicalalloys, where single crystals of the respective tempers are usually not available, theparameters for the crystal plasticity model can be successfully determined by usinga hybrid approach combining micro-compression tests and finite element simulationsincorporating the material model. However, further investigations are required sinceno match between experiment and simulation could be achieved for the specific casewhen a [1 1 1] crystal orientation is aligned with the column axis.

The crystal plasticity model adjusted from micro-compression experiments hasbeen transfered to the oligo-crystalline cantilevers. The finite element simulationspredicted the deformation behavior of these cantilevers both globally, as shown bythe load-displacement curves, and locally, as shown by the DIC measurement ofthe displacement field. Thus, it is possible to use crystal plasticity parametersdetermined from micro-compression experiments to predict the deformation of amicro-cantilever.

Finally, the finite element simulations of the cantilevers were extended by acohesive zone. The initiation stress at the grain boundary was adjusted in termsof the traction-separation law of the cohesive zone. The simulations were moresensitive to the initiation stress for fracture than to the fracture energy. Therefore,the fracture energy was assumed to have a low constant value. This is consistent withthe fact that no propagation of a grain boundary crack could be measured from the

78

experiments. However, such measurements would be required for the determinationof the fracture energy in terms of the cohesive zone model.

The methodology presented within this work can be used to investigate thefracture properties of a specific single grain boundary. The anisotropic plastic de-formation of the grains adjacent to the grain boundary of interest is sufficientlydescribed within this model to determine an initiation stress for a specific grainboundary in terms of a cohesive zone model. This initiation stress is related to theinner structure of the grain boundary and possible effects of strain gradients andboundary conditions at the grain boundary.

In the future such experiments may assist in investigations of the influence ofthe structure of grain boundaries in technical alloys on the fracture behavior [2] andin the research of liquid metal embrittlement of grain boundaries [75, 79]. For suchinvestigations the determination of the fracture energy depending on the charac-ter of the respective grain boundary is important. This requires experiments withmeasurable crack propagation. Possibly, this can be achieved by using trapezoidalcross-sections of the cantilevers which provide increasing crack widths with increas-ing crack lengths. Moreover, the methodology can be extended to investigate theinfluence of shear stresses on the failure of grain boundaries. However, any influencefrom the ions used during the FIB preparation of the specimens on the grain bound-ary properties must be avoided. Hence, for aluminum alloys the gallium source mustbe replaced. A good candidate is a xenon plasma source which is currently underdevelopment.

79

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Vorabveroffentlichung

D Kupka and ET Lilleodden. Mechanical testing of solid-solid interfaces at themicroscale. Experimental Mechanics, 52:649–658, 2012

i

Danksagung

Die vorliegende Arbeit entstand wahrend meiner Tatigkeit als wissenschaftlicherMitarbeiter in der Abteilung Experimentelle Werkstoffmechanik der Helmholtz-Zentrum Geesthacht GmbH.

Herrn Prof. Dr.-Ing. habil. Norbert Huber danke ich fur die Moglichkeit dieseArbeit am Institut fur Werkstoffmechanik am Helmholtz-Zentrum Geesthacht an-fertigen zu durfen. Sein großes Engagement bei der Initiierung und Umsetzung vonForschungsvorhaben werden mir in Erinnerung bleiben, ebenso wie seine Offenheitgegenuber neuen Ideen und deren Umsetzung. Des Weiteren mochte ich Herrn Prof.Dr. rer. nat. Gerold Schneider fur die Ubernahme des Zweitgutachtens dieser Ar-beit danken.

Besonders bedanken mochte ich mich bei Frau Dr. Erica Lilleodden fur die Betreu-ung und ihren personlichen Einsatz.

Herrn Dr. Ingo Scheider und Herrn Dr. Dirk Steglich danke ich fur die Unterstutzungbei der Umsetzung der Simulationen. Bei Frau Julia Hutsch und Frau Petra Fischermochte ich mich fur die Hilfe bei der Erarbeitung der grundlegenden Methoden zurDurchfuhrung der mikro-mechanischen Experimente bedanken. Daruber hinaus binich Herrn Dr. Dirk Schnubel fur die Diskussionen sowie das Korrekturlesen der Ar-beit dankbar.

Mein Dank gilt den ehemaligen und derzeitigen Mitarbeitern der Abteilungen Exper-imentelle Werkstoffmechanik und Simulation von Werkstoff- und Strukturverhaltenfur die angenehme Arbeitsatmosphare und ihre Unterstutzung.

Nicht zuletzt bedanke ich mich bei Herrn Dr. Jurgen Markmann, Herrn Dr. MarkusZiehmer, Anne Groth, Josephin Enz und Stefan Riekher fur die Hilfe bei der Aus-richtung der kleinen Feier nach meiner Promotionsprufung.

Meinen Eltern Wolfgang und Brigitte Kupka danke ich besonders fur die vorbehalt-lose Unterstutzung und den Ruckhalt bei allen Entscheidungen.

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Fracture Experiments of Single Grain Boundaries in Ductile ...

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